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<?xml version="1.0" encoding="utf-8"?>
<feed xmlns="http://www.w3.org/2005/Atom">
<id>https://zy-zzf2000.github.io/</id>
<title>小张的网站</title>
<updated>2020-06-04T16:53:35.678Z</updated>
<generator>https://github.com/jpmonette/feed</generator>
<link rel="alternate" href="https://zy-zzf2000.github.io/"/>
<link rel="self" href="https://zy-zzf2000.github.io/atom.xml"/>
<subtitle>莫道君行早，更有早行人</subtitle>
<logo>https://zy-zzf2000.github.io/images/avatar.png</logo>
<icon>https://zy-zzf2000.github.io/favicon.ico</icon>
<rights>All rights reserved 2020, 小张的网站</rights>
<entry>
<title type="html"><![CDATA[Cramming More Components onto Integrated Circuits]]></title>
<id>https://zy-zzf2000.github.io/post/cramming-more-components-onto-integrated-circuits</id>
<link href="https://zy-zzf2000.github.io/post/cramming-more-components-onto-integrated-circuits">
</link>
<updated>2020-06-04T12:04:49.000Z</updated>
<content type="html"><![CDATA[<p>预测十年之后集成电路的发展情况<br>
With unit cost falling as the number of components per circuit rises, by 1975 economics may dictate squeezing as many as 65 000 components on a single silicon chip.</p>
<p>集成电路的应用会扩展到许多新领域<br>
The future of integrated electronics is the future of electronics itself. The advantages of integration will bring about a proliferation of electronics, pushing this science into many new areas.</p>
<p>具体的例子<br>
Integrated circuits will lead to such wonders as home computers—or at least terminals connected to a central computer—automatic controls for automobiles, and personal portable communications equipment. The electronic wristwatch needs only a display to be feasible today.</p>
<p>最大的潜力在于大型系统的生产<br>
But the biggest potential lies in the production of large systems. In telephone communications, integrated circuits in digital filters will separate channels on multiplex equipment. Integrated circuits will also switch telephone circuits and perform data processing.</p>
<p>对于计算机的影响<br>
Computers will be more powerful, and will be organized in completely different ways. For example, memories built of integrated electronics may be distributed throughout the machine instead of being concentrated in a central unit. In addition, the improved reliability made possible by integrated circuits will allow the construction of larger processing units. Machines similar to those in existence today will be built at lower costs and with faster turnaround.</p>
<p>I. PRESENT AND FUTURE<br>
将集成电路集成化程度提高的几个方法，以及未来的发展趋势是几个方法的融合<br>
By integrated electronics, I mean all the various technologies which are referred to as microelectronics today as well as any additional ones that result in electronics functions supplied to the user as irreducible units. These technologies were first investigated in the late 1950’s. The object was to miniaturize electronics equipment to include increasingly complex electronic functions in limited space with minimum weight. Several approaches evolved, including microassembly techniques for individual components, thin-film structures, and semiconductor integrated circuits.<br>
Each approach evolved rapidly and converged so that each borrowed techniques from another. Many researchers believe the way of the future to be a combination of the various approaches.</p>
<p>The advocates of semiconductor integrated circuitry are already using the improved characteristics of thin-film resistors by applying such films directly to an active semiconductor substrate. Those advocating a technology based upon films are developing sophisticated techniques for the attachment of active semiconductor devices to the passive film arrays.</p>
<p>Both approaches have worked well and are being used in equipment today.</p>
<p>II. THE ESTABLISHMENT<br>
主要叙述了集成电子学相对于包括晶体管在内的传统电子制造的优势,高可靠性，降低了制造以及设计成本，提高性能<br>
Integrated electronics is established today. Its techniques are almost mandatory for new military systems, since the reliability, size, and weight required by some of them is achievable only with integration. Such programs as Apollo, for manned moon flight, have demonstrated the reliability of integrated electronics by showing that complete circuit functions are as free from failure as the best individual transistors</p>
<p>Most companies in the commercial computer field have machines in design or in early production employing integrated electronics. These machines cost less and perform better than those which use “conventional” electronics.</p>
<p>Instruments of various sorts, especially the rapidly increasing numbers employing digital techniques, are starting to use integration because it cuts costs of both manufacture and design.</p>
<p>The use of linear integrated circuitry is still restricted primarily to the military. Such integrated functions are expensive and not available in the variety required to satisfy a major fraction of linear electronics. But the first applications are beginning to appear in commercial electronics, particularly in equipment which needs low-frequency amplifiers of small size.</p>
<p>III. RELIABILITY COUNTS<br>
In almost every case, integrated electronics has demonstrated high reliability. Even at the present level of production—low compared to that of discrete components—it offers reduced systems cost, and in many systems improved performance has been realized.</p>
<p>Integrated electronics will make electronic techniques more generally available throughout all of society, performing many functions that presently are done inadequately by other techniques or not done at all. The principal advantages will be lower costs and greatly simplified design—payoffs<br>
from a ready supply of low-cost functional packages.</p>
<p>For most applications, semiconductor integrated circuits will predominate. Semiconductor devices are the only reasonable candidates presently in existence for the active elements of integrated circuits. Passive semiconductor elements look attractive too, because of their potential for<br>
low cost and high reliability, but they can be used only if precision is not a prime requisite.</p>
<p>Silicon is likely to remain the basic material, although others will be of use in specific applications. For example, gallium arsenide will be important in integrated microwave functions. But silicon will predominate at lower frequencies because of the technology which has already evolved<br>
around it and its oxide, and because it is an abundant and relatively inexpensive starting material.</p>
<p>IV. COSTS AND CURVES<br>
Reduced cost is one of the big attractions of integrated<br>
electronics, and the cost advantage continues to increase<br>
as the technology evolves toward the production of larger<br>
and larger circuit functions on a single semiconductor<br>
substrate. For simple circuits, the cost per component is<br>
nearly inversely proportional to the number of components,<br>
the result of the equivalent piece of semiconductor in<br>
the equivalent package containing more components. But<br>
as components are added, decreased yields more than<br>
compensate for the increased complexity, tending to raise<br>
the cost per component. Thus there is a minimum cost<br>
at any given time in the evolution of the technology. At<br>
present, it is reached when 50 components are used per<br>
circuit. But the minimum is rising rapidly while the entire<br>
cost curve is falling (see graph). If we look ahead five<br>
years, a plot of costs suggests that the minimum cost per<br>
component might be expected in circuits with about 1000<br>
components per circuit (providing such circuit functions<br>
can be produced in moderate quantities). In 1970, the<br>
manufacturing cost per component can be expected to be<br>
only a tenth of the present cost.</p>
<p>The complexity for minimum component costs has increased at a rate of roughly a factor of two per year<br>
(see graph). Certainly over the short term this rate can be<br>
expected to continue, if not to increase. Over the longer<br>
term, the rate of increase is a bit more uncertain, although<br>
there is no reason to believe it will not remain nearly<br>
constant for at least ten years. That means by 1975, the<br>
number of components per integrated circuit for minimum<br>
cost will be 65 000.</p>
<p>I believe that such a large circuit can be built on a single<br>
wafer</p>
<p>V. TWO-MIL SQUARES<br>
With the dimensional tolerances already being employed<br>
in integrated circuits, isolated high-performance transistors<br>
can be built on centers two-thousandths of an inch apart.<br>
Such a two-mil square can also contain several kilohms<br>
of resistance or a few diodes. This allows at least 500<br>
components per linear inch or a quarter million per square<br>
inch. Thus, 65 000 components need occupy only about<br>
one-fourth a square inch.</p>
<p>On the silicon wafer currently used, usually an inch or<br>
more in diameter, there is ample room for such a structure if<br>
the components can be closely packed with no space wasted<br>
for interconnection patterns. This is realistic, since efforts to<br>
achieve a level of complexity above the presently available<br>
integrated circuits are already under way using multilayer<br>
metallization patterns separated by dielectric films. Such a<br>
density of components can be achieved by present optical<br>
techniques and does not require the more exotic techniques,<br>
such as electron beam operations, which are being studied<br>
to make even smaller structures.</p>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[霍夫丁定理证明]]></title>
<id>https://zy-zzf2000.github.io/post/huo-fu-ding-ding-li-zheng-ming</id>
<link href="https://zy-zzf2000.github.io/post/huo-fu-ding-ding-li-zheng-ming">
</link>
<updated>2020-01-24T08:00:36.000Z</updated>
<content type="html"><![CDATA[<blockquote>
<p>资料来源于yzg大佬，涵盖了马尔科夫不等式、切比雪夫不等式、切诺夫界、霍夫丁引</p>
</blockquote>
<p>一、<br>
<img src="https://zy-zzf2000.github.io//post-images/1579852927103.jpg" alt="" loading="lazy"></p>
<p>二、<br>
<img src="https://zy-zzf2000.github.io//post-images/1579852949581.jpg" alt="" loading="lazy"></p>
<p>三、<br>
<img src="https://zy-zzf2000.github.io//post-images/1579852957653.jpg" alt="" loading="lazy"></p>
<p>四、<br>
<img src="https://zy-zzf2000.github.io//post-images/1579852968404.jpg" alt="" loading="lazy"></p>
<hr>
<p>由于zy比较懒，没有进行整理...下面对一开始没有看懂的地方进行一定的补充：</p>
<h4 id="一矩量母函数">（一）矩量母函数</h4>
<p>对于随机变量ξ，我们称exp(tξ)的数学期望为随机变量ξ的矩量母函数，记作mξ(t)=E(exp(tξ))</p>
<hr>
<h4 id="二凸函数的一般性质">（二）凸函数的一般性质</h4>
<p>对于任意的<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">λ</span></span></span></span> 属于（0，1），若<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> ,<span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>x</mi><mn>2</mn></msub></mrow><annotation encoding="application/x-tex">x_2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> 在某凸函数的定义域内，则有下列等式成立：</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>f</mi><mo>(</mo><mi>λ</mi><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo>)</mo><msub><mi>x</mi><mn>2</mn></msub><mo>)</mo><mo><</mo><mo>=</mo><mi>λ</mi><mi>f</mi><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo>)</mo><mo>+</mo><mo>(</mo><mn>1</mn><mo>−</mo><mi>λ</mi><mo>)</mo><mi>f</mi><mo>(</mo><msub><mi>x</mi><mn>2</mn></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">f(\lambda x_1+(1-\lambda)x_2)<=\lambda f(x_1)+(1-\lambda)f(x_2)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">λ</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">λ</span><span class="mclose">)</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><</span></span><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">λ</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">1</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">λ</span><span class="mclose">)</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p>
<hr>
<h4 id="三图片3中利用切诺边界">(三)图片3中利用切诺边界</h4>
<p>在利用切诺边界时，同时还利用了Z1，Z2...Zn是相互独立的变量。</p>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[朴素贝叶斯法补充]]></title>
<id>https://zy-zzf2000.github.io/post/-Bayes</id>
<link href="https://zy-zzf2000.github.io/post/-Bayes">
</link>
<updated>2020-01-20T03:15:44.000Z</updated>
<content type="html"><![CDATA[<h2 id="一-如何理解条件概率分布具有指数级别的参数">一、如何理解条件概率分布具有指数级别的参数</h2>
<p>我个人比较能理解的一种说法是：这里的概率分布学习，是以统计频数的方式进行的，因此，从模型的角度来看，模型参数的个数应当与事件{X=x|Y=Ck}成正比（或相等？），即共有：</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>K</mi><munderover><mo>∏</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>n</mi></munderover><msub><mi>S</mi><mi>j</mi></msub></mrow><annotation encoding="application/x-tex">K\prod_{j=1}^{n}{S_j}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0651740000000007em;vertical-align:-1.4137769999999998em;"></span><span class="mord mathdefault" style="margin-right:0.07153em;">K</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.6513970000000007em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∏</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">n</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.4137769999999998em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.311664em;"><span style="top:-2.5500000000000003em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.05724em;">j</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.286108em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>式中，K为Y的可能值的个数，Sj为x(j)可能的取值数。</p>
<hr>
<h2 id="二-基础知识回顾">二、基础知识回顾</h2>
<h4 id="一先验概率与后验概率">（一）先验概率与后验概率</h4>
<p>先给出简洁的定义：</p>
<p>先验概率：根据以往经验分析得到的概率</p>
<p>后验概率：在事件发生后,根据事件发生的情况,对先验概率进行的一种修正</p>
<p>下面有两个具体的例子：</p>
<blockquote>
<p>情景一：</p>
<p>现在发生一件事：有人揭开了 C 碗，发现 C 碗下面没有蛋。此时再问：鸡蛋在 A 碗下面的概率是多少？答曰 1/2。注意，由于有“揭开C碗发现鸡蛋不在C碗下面”这个新情况，对于“鸡蛋在 A 碗下面”这件事的主观概率由原来的 1/3 上升到了1/2。这里的先验概率就是 1/3，后验概率是 1/2。<br>
也就是说“先”和“后”是相对于引起主观概率变化的那个新情况而言的。</p>
<p>情景二：</p>
<p>“天不会下雨的，历史上这里下雨的概率是20%”----先验概率<br>
“但阴云漠漠时，下雨的概率是80%”----后验概率</p>
</blockquote>
<p>在朴素贝叶斯法中，先验概率即P（Y=Ck），后验概率即P（Y=Ck|X=x）</p>
<hr>
<h4 id="二极大似然估计">（二）极大似然估计</h4>
<p>所谓似然，就是通过n个随机变量的实验结果来估计随机变量的参数。其中使得当前实验结果出现概率最大的取值就被称作是极大似然估计。<a href="https://www.zhihu.com/question/24124998/answer/242682386">具体可以参考这里</a></p>
<hr>
<h4 id="三拉格朗日乘子法">（三）拉格朗日乘子法</h4>
<p>拉格朗日乘子法是是一种寻找变量受一个或多个条件所限制的多元函数的极值的方法。在这里，我仅仅从二维层面直观上对其进行理解：<br>
<img src="https://zy-zzf2000.github.io//post-images/1579582388602.jpg" alt="" loading="lazy"><br>
如图所示，虚线代表了待求极值的函数f（x，y）的等高线，红色曲线代表了约束函数g（x，y）=0。在施加约束条件之前，我们可以在平面内任意取点，得到不同的函数值；而在施加约束条件之后，我们就只能取g（x，y）上的点。不难理解，我们所要求的极值点一定是g（x，y）=0和某条等高线的切点。因为只要f（x，y）的一条等高线与g（x，y）=0相交，那么就一定存在取值更小（或更大）的等高线与g（x，y）=0相交，因此极值一定在切点处取得。由于两条曲线f（x，y）=C（C即为我们所求的极值）与g（x，y）=0在极值点相切，那么它们的法向量必然只相差一个常数，即有：</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">∇</mi><mi>f</mi><mi mathvariant="normal">（</mi><mi>x</mi><mi mathvariant="normal">，</mi><mi>y</mi><mi mathvariant="normal">）</mi><mo>=</mo><mo>−</mo><mi>λ</mi><mi mathvariant="normal">∇</mi><mi>g</mi><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>)</mo></mrow><annotation encoding="application/x-tex">\nabla f（x，y）=-\lambda\nabla g(x,y)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">∇</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mord cjk_fallback">（</span><span class="mord mathdefault">x</span><span class="mord cjk_fallback">，</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mord cjk_fallback">）</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mord mathdefault">λ</span><span class="mord">∇</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span></span></p>
<p>经过移项，可以得到：</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi mathvariant="normal">∇</mi><mo>(</mo><mi>f</mi><mo>(</mo><mi>x</mi><mi mathvariant="normal">，</mi><mi>y</mi><mo>)</mo><mo>+</mo><mi>λ</mi><mi>g</mi><mo>(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo>)</mo><mo>)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\nabla(f(x，y)+\lambda g(x,y))=0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">∇</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mord cjk_fallback">，</span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">λ</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord mathdefault">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></p>
<p>在使用过程中，我们只需要令：</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>F</mi><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo><mo>+</mo><mi>λ</mi><mi>g</mi><mo>(</mo><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><msub><mi>x</mi><mi>n</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">F(x_1,x_2,...,x_n)=f(x_1,x_2,...x_n)+\lambda g(x_1,x_2,...x_n)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault" style="margin-right:0.10764em;">f</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">λ</span><span class="mord mathdefault" style="margin-right:0.03588em;">g</span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p>
<h2 id="然后分别用f对x1x2xn求偏导再联立约束条件就能求出极值点">然后分别用F对x1，x2...,xn求偏导，再联立约束条件就能求出极值点。</h2>
<h2 id="三-先验概率的推导">三、先验概率的推导</h2>
<figure data-type="image" tabindex="1"><img src="https://zy-zzf2000.github.io//post-images/1579582558699.jpg" alt="" loading="lazy"></figure>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[K近邻补充]]></title>
<id>https://zy-zzf2000.github.io/post/k-jin-lin-bu-chong</id>
<link href="https://zy-zzf2000.github.io/post/k-jin-lin-bu-chong">
</link>
<updated>2020-01-18T10:38:16.000Z</updated>
<content type="html"><![CDATA[<h2 id="一-单元概念">一、单元概念</h2>
<blockquote>
<p>书p50:对每个训练实例点xi，距离该点比其他点更近的所有点组成一个区域，叫做单元（cell）。</p>
</blockquote>
<p>对于这句话个人的理解是，训练实例xi的单元是这样的点组成的集合，即集合中任意一个点到训练集中其他点的距离都要大于这个点到xi的距离。</p>
<h2 id="二-kd树">二、kd树</h2>
<h4 id="1kd树的概念">（1）kd树的概念</h4>
<p>kd树即k-dimensional tree，其k代表的是训练实例的维数，与k近邻法的k意义不同。本质上是二叉查找树在k维的扩展。其节点对应于一个n维的超矩形区域，储存的是一个子区域的划分。</p>
<h4 id="2维数的选择">（2）维数的选择</h4>
<p>在书中，维数的选择是从1，2，3...循环选择的，即对于深度为j的结点，选择第l=j%k+1作为切分的坐标轴。除此之外，还有一种被称作是 最大方差法 的维度选择法，它的基本做法是每次在选择维度时，我们总是倾向于选择那些方差最大的的维度。这样做的基本思想是，数据在某个维度越分散，则越易于区分；越密集，则区分就越为困难。</p>
<h4 id="3近邻搜索">（3）近邻搜索</h4>
<p>近邻搜索首先找到包含待预测点x的区域的叶节点，并以此叶结点作为x的近似最近邻。之后不断递归，找当前近似最近邻的另一个兄弟节点所代表的区域内是否与以 x为圆心，r=||x-当前最近邻|| 的圆相交，若相交部分有点，则以该点为新的近似最近邻；若无则返回上一次继续递归。</p>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[感知机补充]]></title>
<id>https://zy-zzf2000.github.io/post/preceptron</id>
<link href="https://zy-zzf2000.github.io/post/preceptron">
</link>
<updated>2020-01-17T02:09:18.000Z</updated>
<content type="html"><![CDATA[<p>##一、超平面</p>
<h3 id="1什么是超平面">（1）什么是超平面</h3>
<blockquote>
<p>超平面是n维欧氏空间中余维度等于一的线性子空间，也就是必须是(n-1)维度。</p>
</blockquote>
<p>例如在一维数轴中，点可以将数轴划分为两个部分，在二维平面中，一条直线可以将平面区域划分为两个部分，在三维空间中，一个平面可以将空间划分为两个不同的部分...扩展到n维，则有超平面的概念。</p>
<h3 id="2超平面的方程形式">（2）超平面的方程形式</h3>
<p>由直线方程ax + by + c = 0</p>
<p>平面方程ax + by + cz + d = 0</p>
<p>不难类推出超平面方程</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>a</mi><mn>1</mn></msub><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><msub><mi>a</mi><mn>2</mn></msub><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo>+</mo><msub><mi>a</mi><mi>n</mi></msub><msub><mi>x</mi><mi>n</mi></msub><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">a_1x_1+a_2x_2+...+a_nx_n+b=0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></p>
<p>在这里，我们令</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>w</mi><mo>=</mo><mi mathvariant="normal">（</mi><msub><mi>a</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>a</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>a</mi><mi>n</mi></msub><mo>)</mo></mrow><annotation encoding="application/x-tex">w=（a_1,a_2,...,a_n)
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord cjk_fallback">（</span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">a</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>x</mi><mo>=</mo><mi mathvariant="normal">（</mi><msub><mi>x</mi><mn>1</mn></msub><mo separator="true">,</mo><msub><mi>x</mi><mn>2</mn></msub><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>x</mi><mi>n</mi></msub><msup><mo>)</mo><mi>T</mi></msup></mrow><annotation encoding="application/x-tex"> x=（x_1,x_2,...,x_n)^T
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.1413309999999999em;vertical-align:-0.25em;"></span><span class="mord cjk_fallback">（</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose"><span class="mclose">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.13889em;">T</span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>于是，上述方程就可以被表示为：</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>w</mi><mo separator="true">⋅</mo><mi>x</mi><mo>+</mo><mi>b</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">w·x+b=0
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">x</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></span></p>
<p>其中，w应当是该超平面的一个法向量。</p>
<p>由此我们可以看出，所谓感知机的分类器，就是要找到这样一个超平面，使得数据集的正实例点和负实例点被完全正确的划分到该超平面的两侧。</p>
<h3 id="3超平面距离公式">（3）超平面距离公式</h3>
<figure data-type="image" tabindex="1"><img src="https://zy-zzf2000.github.io//post-images/1579250261999.png" alt="" loading="lazy"></figure>
<p>如图，E为超平面内一点，F为超平面外一点，则F到平面距离为：</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∣</mi><mi>w</mi><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><mi>b</mi><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>w</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">d=\frac{|wx_0+b|}{||w||}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">d</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mord">∣</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">b</span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>下面给出推导过程：</p>
<p>首先由余弦公式：</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi><mi>o</mi><mi>s</mi><mi>θ</mi><mo>=</mo><mfrac><mrow><mi>w</mi><mo separator="true">⋅</mo><mi>x</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>w</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>×</mo><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">cos\theta=\frac{w·x}{||w||×||x||}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault">o</span><span class="mord mathdefault">s</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.0574500000000002em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.12145em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mord">∣</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">×</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">∣</span><span class="mord">∣</span><span class="mord mathdefault">x</span><span class="mord">∣</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>推出:</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>c</mi><mi>o</mi><mi>s</mi><mi>θ</mi><mo separator="true">⋅</mo><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>=</mo><mfrac><mrow><mi>w</mi><mo separator="true">⋅</mo><mi>x</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>w</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>−</mo><mi>b</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>w</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">cos\theta ·||x||=\frac{w·x}{||w||}=\frac{-b}{||w||}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault">o</span><span class="mord mathdefault">s</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mord">∣</span><span class="mord mathdefault">x</span><span class="mord">∣</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.0574500000000002em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.12145em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mord">∣</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">x</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.30744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mord">∣</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">−</span><span class="mord mathdefault">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>由此我们可以看出，超平面上任意一点的向量在法向量w上投影的长度都等于一个定值，而由图易知，向量OF在法向量上投影长度为</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>L</mi><mo>=</mo><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>O</mi><mi>F</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo separator="true">⋅</mo><mi>c</mi><mi>o</mi><mi>s</mi><msup><mi>θ</mi><mo mathvariant="normal">′</mo></msup><mo>=</mo><mfrac><mrow><mi>w</mi><mo separator="true">⋅</mo><mi>O</mi><mi>F</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>w</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">L=||OF||·cos\theta'=\frac{w·OF}{||w||}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">L</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.051892em;vertical-align:-0.25em;"></span><span class="mord">∣</span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mord">∣</span><span class="mord">∣</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault">o</span><span class="mord mathdefault">s</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.801892em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.29633em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.36033em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mord">∣</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<p>所以有：</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>d</mi><mo>=</mo><mi>c</mi><mi>o</mi><mi>s</mi><mi>θ</mi><mo separator="true">⋅</mo><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>x</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mo>−</mo><mi>L</mi><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∣</mi><mo>−</mo><mi>b</mi><mo>−</mo><mi>w</mi><mo separator="true">⋅</mo><mi>O</mi><mi>F</mi><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>w</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∣</mi><mi>b</mi><mo>+</mo><mi>w</mi><mo separator="true">⋅</mo><msub><mi>x</mi><mn>0</mn></msub><mi mathvariant="normal">∣</mi></mrow><mrow><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi><mi>w</mi><mi mathvariant="normal">∣</mi><mi mathvariant="normal">∣</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">d=cos\theta ·||x||-L=\frac{|-b-w·OF|}{||w||}=\frac{|b+w·x_0|}{||w||}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathdefault">d</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">c</span><span class="mord mathdefault">o</span><span class="mord mathdefault">s</span><span class="mord mathdefault" style="margin-right:0.02778em;">θ</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">∣</span><span class="mord">∣</span><span class="mord mathdefault">x</span><span class="mord">∣</span><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathdefault">L</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mord">∣</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault" style="margin-right:0.02778em;">O</span><span class="mord mathdefault" style="margin-right:0.13889em;">F</span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord">∣</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mord">∣</span><span class="mord">∣</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">∣</span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">⋅</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">∣</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span></p>
<hr>
<hr>
<h2 id="二-损失函数l梯度推导">二、损失函数L梯度推导</h2>
<p>损失函数的梯度公式为：</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="normal">∇</mi><mi>w</mi></msub><mi>L</mi><mo>(</mo><mi>w</mi><mo separator="true">,</mo><mi>b</mi><mo>)</mo><mo>=</mo><mo>−</mo><mo>∑</mo><mrow><msub><mi>y</mi><mi>i</mi></msub><msub><mi>x</mi><mi>i</mi></msub></mrow></mrow><annotation encoding="application/x-tex">\nabla_wL(w,b)=-\sum{y_ix_i}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">L</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.6000100000000002em;vertical-align:-0.55001em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-symbol large-op" style="position:relative;top:-0.000004999999999977245em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="normal">∇</mi><mi>b</mi></msub><mi>L</mi><mo>(</mo><mi>w</mi><mo separator="true">,</mo><mi>b</mi><mo>)</mo><mo>=</mo><mo>−</mo><mo>∑</mo><msub><mi>y</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\nabla_bL(w,b)=-\sum{y_i}
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">L</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.6000100000000002em;vertical-align:-0.55001em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-symbol large-op" style="position:relative;top:-0.000004999999999977245em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></span></p>
<p>公式中的点属于误分类点集合M。</p>
<p>下面对该公式进行推导：</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>L</mi><mo>(</mo><mi>w</mi><mo separator="true">,</mo><mi>b</mi><mo>)</mo><mo>=</mo><mo>−</mo><mo>∑</mo><msub><mi>y</mi><mi>i</mi></msub><mo>(</mo><mi>w</mi><mo>⋅</mo><msub><mi>x</mi><mi>i</mi></msub><mo>+</mo><mi>b</mi><mo>)</mo><mo>=</mo><mo>−</mo><mo>[</mo><msub><mi>y</mi><mn>1</mn></msub><mo>(</mo><mi>w</mi><mo>⋅</mo><msub><mi>x</mi><mn>1</mn></msub><mo>+</mo><mi>b</mi><mo>)</mo><mo>+</mo><msub><mi>y</mi><mn>2</mn></msub><mo>(</mo><mi>w</mi><mo>⋅</mo><msub><mi>x</mi><mn>2</mn></msub><mo>+</mo><mi>b</mi><mo>)</mo><mo>+</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo>+</mo><msub><mi>y</mi><mi>n</mi></msub><mo>(</mo><mi>w</mi><mo>⋅</mo><msub><mi>x</mi><mi>n</mi></msub><mo>+</mo><mi>b</mi><mo>)</mo><mo>]</mo></mrow><annotation encoding="application/x-tex">L(w,b)=-\sum y_i(w\cdot x_i+b)=-[y_1(w\cdot x_1+b)+y_2(w\cdot x_2+b)+...+y_n(w\cdot x_n+b)]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">L</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.6000100000000002em;vertical-align:-0.55001em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-symbol large-op" style="position:relative;top:-0.000004999999999977245em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.73333em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span><span class="mclose">]</span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>=</mo><mo>−</mo><mo>[</mo><msub><mi>y</mi><mn>1</mn></msub><mo>(</mo><msup><mi>w</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>⋅</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup><mo>+</mo><msup><mi>w</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msup><mo>⋅</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo>+</mo><msup><mi>w</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>⋅</mo><msubsup><mi>x</mi><mn>1</mn><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msubsup><mo>+</mo><mi>b</mi><mo>)</mo><mo>+</mo></mrow><annotation encoding="application/x-tex">=-[y_1(w^{(1)}\cdot x_1^{(1)} + w^{(2)}\cdot x_1^{(2)} +...+w^{(n)}\cdot x_1^{(n)} +b)+
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord">−</span><span class="mopen">[</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.266308em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.266308em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.938em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">2</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.266308em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">2</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.266308em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.938em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.266308em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.266308em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span><span class="mord">+</span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>=</mo><msub><mi>y</mi><mn>2</mn></msub><mo>(</mo><msup><mi>w</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>⋅</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup><mo>+</mo><msup><mi>w</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msup><mo>⋅</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo>+</mo><msup><mi>w</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>⋅</mo><msubsup><mi>x</mi><mi>n</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msubsup><mo>+</mo><mi>b</mi><mo>)</mo><mo>+</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi></mrow><annotation encoding="application/x-tex">=y_2(w^{(1)}\cdot x_2^{(1)} + w^{(2)}\cdot x_2^{(2)} +...+w^{(n)}\cdot x_n^{(n)}+b )+...
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.266308em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.266308em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.938em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">2</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.266308em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">2</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.266308em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.938em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.10556em;vertical-align:0em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>=</mo><mo>+</mo><msub><mi>y</mi><mi>n</mi></msub><mo>(</mo><msup><mi>w</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>⋅</mo><msubsup><mi>x</mi><mi>n</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup><mo>+</mo><msup><mi>w</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msup><mo>⋅</mo><msubsup><mi>x</mi><mn>2</mn><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msubsup><mo>+</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo>+</mo><msup><mi>w</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>⋅</mo><msubsup><mi>x</mi><mi>n</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msubsup><mo>+</mo><mi>b</mi><mo>)</mo><mo>]</mo></mrow><annotation encoding="application/x-tex">=+y_n(w^{(1)}\cdot x_n^{(1)} + w^{(2)}\cdot x_2^{(2)} +...+w^{(n)}\cdot x_n^{(n)} +b)]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.188em;vertical-align:-0.25em;"></span><span class="mord">+</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.938em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">2</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.311108em;vertical-align:-0.266308em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.433692em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">2</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.266308em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.938em;vertical-align:0em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.185em;vertical-align:-0.247em;"></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-2.4530000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">n</span></span></span><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.247em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span><span class="mclose">]</span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>=</mo><mo>−</mo><mo>[</mo><mo>(</mo><mo>∑</mo><msub><mi>y</mi><mi>i</mi></msub><msubsup><mi>x</mi><mi>i</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup><mo>)</mo><msup><mi>w</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup><mo>+</mo><mo>(</mo><mo>∑</mo><msub><mi>y</mi><mi>i</mi></msub><msubsup><mi>x</mi><mi>i</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msubsup><mo>)</mo><msup><mi>w</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msup><mo>+</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo>+</mo><mo>(</mo><mo>∑</mo><msub><mi>y</mi><mi>i</mi></msub><msubsup><mi>x</mi><mi>i</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msubsup><mo>)</mo><msup><mi>w</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mo>+</mo><mi>b</mi><mo>∑</mo><msub><mi>y</mi><mi>i</mi></msub><mo>]</mo></mrow><annotation encoding="application/x-tex">=-[(\sum y_ix_i^{(1)})w^{(1)}+(\sum y_ix_i^{(2)})w^{(2)}+...+(\sum y_ix_i^{(n)})w^{(n)}+b\sum y_i]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.6000100000000002em;vertical-align:-0.55001em;"></span><span class="mord">−</span><span class="mopen">[</span><span class="mopen">(</span><span class="mop op-symbol large-op" style="position:relative;top:-0.000004999999999977245em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.6000100000000002em;vertical-align:-0.55001em;"></span><span class="mopen">(</span><span class="mop op-symbol large-op" style="position:relative;top:-0.000004999999999977245em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">2</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">2</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.6000100000000002em;vertical-align:-0.55001em;"></span><span class="mopen">(</span><span class="mop op-symbol large-op" style="position:relative;top:-0.000004999999999977245em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.938em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.6000100000000002em;vertical-align:-0.55001em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-symbol large-op" style="position:relative;top:-0.000004999999999977245em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span></p>
<p>再分别对w、b求偏导（其中w为对向量求偏导数）</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="normal">∇</mi><mi>w</mi></msub><mi>L</mi><mo>(</mo><mi>w</mi><mo separator="true">,</mo><mi>b</mi><mo>)</mo><mo>=</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>L</mi><mo>(</mo><mi>w</mi><mo separator="true">,</mo><mi>b</mi><mo>)</mo></mrow><mrow><mi mathvariant="normal">∂</mi><mi>w</mi></mrow></mfrac><mo>=</mo><mo>[</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>L</mi><mo>(</mo><mi>w</mi><mo separator="true">,</mo><mi>b</mi><mo>)</mo></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>w</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msup></mrow></mfrac><mo separator="true">,</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>L</mi><mo>(</mo><mi>w</mi><mo separator="true">,</mo><mi>b</mi><mo>)</mo></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>w</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msup></mrow></mfrac><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><mfrac><mrow><mi mathvariant="normal">∂</mi><mi>L</mi><mo>(</mo><mi>w</mi><mo separator="true">,</mo><mi>b</mi><mo>)</mo></mrow><mrow><mi mathvariant="normal">∂</mi><msup><mi>w</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup></mrow></mfrac><mo>]</mo><mo>=</mo><mo>−</mo><mo>[</mo><mo>∑</mo><msub><mi>y</mi><mi>i</mi></msub><msubsup><mi>x</mi><mi>i</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></msubsup><mo separator="true">,</mo><msub><mi>y</mi><mi>i</mi></msub><msubsup><mi>x</mi><mi>i</mi><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></msubsup><mo separator="true">,</mo><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mi mathvariant="normal">.</mi><mo separator="true">,</mo><msub><mi>y</mi><mi>i</mi></msub><msubsup><mi>x</mi><mi>i</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msubsup><mo>]</mo></mrow><annotation encoding="application/x-tex">\nabla_wL(w,b)=\frac{∂L(w,b)}{∂w}=[\frac{∂L(w,b)}{∂w^{(1)}},\frac{∂L(w,b)}{∂w^{(2)}},...,\frac{∂L(w,b)}{∂w^{(n)}}]=-[\sum y_ix_i^{(1)},y_ix_i^{(2)},...,y_ix_i^{(n)}]
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.151392em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight" style="margin-right:0.02691em;">w</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">L</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathdefault">L</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.1310000000000002em;vertical-align:-0.704em;"></span><span class="mopen">[</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.2960000000000003em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.814em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathdefault">L</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.704em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.2960000000000003em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.814em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">2</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathdefault">L</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.704em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.2960000000000003em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord"><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.814em;"><span style="top:-2.989em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord" style="margin-right:0.05556em;">∂</span><span class="mord mathdefault">L</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.704em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mclose">]</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.6000100000000002em;vertical-align:-0.55001em;"></span><span class="mord">−</span><span class="mopen">[</span><span class="mop op-symbol large-op" style="position:relative;top:-0.000004999999999977245em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mtight">2</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord">.</span><span class="mord">.</span><span class="mord">.</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.0448em;"><span style="top:-2.4231360000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span><span style="top:-3.2198em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mopen mtight">(</span><span class="mord mathdefault mtight">n</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.276864em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mo>=</mo><mo>−</mo><mo>∑</mo><msub><mi>y</mi><mi>i</mi></msub><msub><mi>x</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">=-\sum y_ix_i
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.6000100000000002em;vertical-align:-0.55001em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-symbol large-op" style="position:relative;top:-0.000004999999999977245em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi mathvariant="normal">∇</mi><mi>b</mi></msub><mi>L</mi><mo>(</mo><mi>w</mi><mo separator="true">,</mo><mi>b</mi><mo>)</mo><mo>=</mo><mo>−</mo><mo>∑</mo><msub><mi>y</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\nabla_bL(w,b)=-\sum y_i
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord">∇</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">b</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault">L</span><span class="mopen">(</span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.6000100000000002em;vertical-align:-0.55001em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-symbol large-op" style="position:relative;top:-0.000004999999999977245em;">∑</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<hr>
<hr>
<h2 id="三-感知机对偶形式">三、感知机对偶形式</h2>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>w</mi><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>α</mi><mi>i</mi></msub><msub><mi>y</mi><mi>i</mi></msub><msub><mi>x</mi><mi>i</mi></msub><mo separator="true">,</mo><mi>b</mi><mo>=</mo><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><msub><mi>α</mi><mi>i</mi></msub><msub><mi>y</mi><mi>i</mi></msub></mrow><annotation encoding="application/x-tex">w=\sum_{i=1}^{N}\alpha_iy_ix_i,b=\sum_{i=1}^{N}\alpha_iy_i
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathdefault" style="margin-right:0.02691em;">w</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.106005em;vertical-align:-1.277669em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathdefault">b</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.106005em;vertical-align:-1.277669em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.8283360000000002em;"><span style="top:-1.872331em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight">i</span><span class="mrel mtight">=</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.050005em;"><span class="pstrut" style="height:3.05em;"></span><span><span class="mop op-symbol large-op">∑</span></span></span><span style="top:-4.3000050000000005em;margin-left:0em;"><span class="pstrut" style="height:3.05em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathdefault mtight" style="margin-right:0.10903em;">N</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:1.277669em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.03588em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></span></p>
<p>式中</p>
<p class='katex-block'><span class="katex-display"><span class="katex"><span class="katex-mathml"><math><semantics><mrow><msub><mi>α</mi><mi>i</mi></msub><mo>=</mo><msub><mi>n</mi><mi>i</mi></msub><mi>η</mi></mrow><annotation encoding="application/x-tex">\alpha_i=n_i\eta
</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathdefault" style="margin-right:0.0037em;">α</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:-0.0037em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathdefault">n</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.31166399999999994em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathdefault mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord mathdefault" style="margin-right:0.03588em;">η</span></span></span></span></span></p>
<p>ni的意义是第i个点在参数更新迭代过程中被使用的次数，<em>η</em>是步长。</p>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[Sonnet 18]]></title>
<id>https://zy-zzf2000.github.io/post/sonnet-18</id>
<link href="https://zy-zzf2000.github.io/post/sonnet-18">
</link>
<updated>2019-11-04T12:24:59.000Z</updated>
<content type="html"><![CDATA[<p>to be continue...</p>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[赤壁赋]]></title>
<id>https://zy-zzf2000.github.io/post/chibifu</id>
<link href="https://zy-zzf2000.github.io/post/chibifu">
</link>
<updated>2019-11-02T06:04:44.000Z</updated>
<content type="html"><![CDATA[<p>壬戌之秋，七月既望，苏子与客泛舟游于赤壁之下。清风徐来，水波不兴。举酒属客，诵明月之诗，歌窈窕之章。少焉，月出于东山之上，徘徊于斗牛之间。白露横江，水光接天。纵一苇之所如，凌万顷之茫然。浩浩乎如冯虚御风，而不知其所止；飘飘乎如遗世独立，羽化而登仙。</p>
<p>于是饮酒乐甚，扣舷而歌之。歌曰：“桂棹兮兰桨，击空明兮溯流光。渺渺兮予怀，望美人兮天一方。”客有吹洞箫者，倚歌而和之。其声呜呜然，如怨如慕，如泣如诉，余音袅袅，不绝如缕。舞幽壑之潜蛟，泣孤舟之嫠妇。</p>
<p>苏子愀然，正襟危坐而问客曰：“何为其然也？”客曰：“月明星稀，乌鹊南飞，此非曹孟德之诗乎？西望夏口，东望武昌，山川相缪，郁乎苍苍，此非孟德之困于周郎者乎？方其破荆州，下江陵，顺流而东也，舳舻千里，旌旗蔽空，酾酒临江，横槊赋诗，固一世之雄也，而今安在哉？况吾与子渔樵于江渚之上，侣鱼虾而友麋鹿，驾一叶之扁舟，举匏樽以相属。<strong>寄蜉蝣于天地，渺沧海之一粟。哀吾生之须臾，羡长江之无穷。挟飞仙以遨游，抱明月而长终。知不可乎骤得，托遗响于悲风。”</strong></p>
<p>苏子曰：“客亦知夫水与月乎？逝者如斯，而未尝往也；盈虚者如彼，而卒莫消长也。盖将自其变者而观之，则天地曾不能以一瞬；自其不变者而观之，则物与我皆无尽也，而又何羡乎!且夫天地之间，物各有主,苟非吾之所有，虽一毫而莫取。惟江上之清风，与山间之明月，耳得之而为声，目遇之而成色，取之无禁，用之不竭，是造物者之无尽藏也，而吾与子之所共适。”</p>
<p>客喜而笑，洗盏更酌。肴核既尽，杯盘狼籍。相与枕藉乎舟中，不知东方之既白。</p>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[散列]]></title>
<id>https://zy-zzf2000.github.io/post/san-lie</id>
<link href="https://zy-zzf2000.github.io/post/san-lie">
</link>
<updated>2019-10-31T04:54:14.000Z</updated>
<content type="html"><![CDATA[<p>待续...</p>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[KMP算法-从PMT角度理解]]></title>
<id>https://zy-zzf2000.github.io/post/KMP</id>
<link href="https://zy-zzf2000.github.io/post/KMP">
</link>
<updated>2019-10-19T14:43:45.000Z</updated>
<content type="html"><![CDATA[<blockquote>
<p>在最近的课堂上，老师讲了KMP算法，但总感觉老师的推理过程有些不连贯，所以决定自己梳理一遍KMP算法</p>
</blockquote>
<hr>
<h1 id="1-next数组是什么">1、next数组是什么</h1>
<p>KMP算法的核心就是next数组。在KMP算法中，模式字符串的每一位均对应一个next值，即当主字符串与模式字符串在这一位发生失配时，模式字符串的指针不用回溯到0，只需回溯到next值即可。理解next数组是理解KMP算法的关键。</p>
<hr>
<h1 id="2-next数组的由来">2、next数组的由来</h1>
<p>在课本中，KMP算法是由BF算法改进而来的。但我个人认为这样并不好理解（其实是我当初看的时候没看懂，结果上课又没听懂）。在网上查阅之后，发现从PMT数组的角度似乎能更好的理解next数组。</p>
<p>PMT数组，即部分匹配表(Partial Match Table)，与next数组一样，对于模式字符串的每一位，它都有一个对应的值，如下图：<br>
<img src="https://zy-zzf2000.github.io//post-images/1572084892998.jpg" alt="" loading="lazy"><br>
通俗的说，next数组里的值就是模式字符串当前位置子串中，前缀与后缀最大重复元素的长度。这样说也许还是难以理解，让我们用上图来举几个例子。<br>
可以看到，PMT【1】=0，这是因为对于字串“ab”，它的前缀只有‘a’，后缀只有‘b’，因此没有重复部分；PMT【2】=1，这是因为对于字串“aba”，它的前缀有“a”、“ab”，后缀有“ba”、“a”，其中最大重复元素位“a”，因此最大重复元素的长度=PMT值=1。以此类推...</p>
<p>PMT数组有什么用呢？PMT数组可以用来帮我们省去一些不必要的比较。如下图所示：<br>
<img src="https://zy-zzf2000.github.io//post-images/1572085486261.jpg" alt="" loading="lazy"><br>
在图（a）中，主串与模式字符串即将发生失配。由PMT数组我们知道，对于模式字符串的子串“ababab”（即指针j前的字串），它的PMT值为4（PMT【j-1】），也就是说，它的前缀与后缀有4位是重复的。而主串与模式字符串失配前的每一位都是一样的，于是我们可以得出，主串的i指针之前的4位，相当于“ababab”的后缀，而模式字符串的前四位，相当于“ababab”的前缀，他们也是相等的。既然模式字符串的前四位已经相等了，那么我们在下一次进行比较的时候，就没有必要再对它们进行比较了，如图（b）所示，我们保持i指针不变，直接将j指针移动到第五位，直接从模式字符串的第五位开始比较。</p>
<p>从上面我们可以看出，如果再比较时，考虑PMT数组，那么每当发生失配时，我们的j指针就不必回溯到0，只需要回溯到PMT【j-1】所指示的位置。为了编程方便，我们将PMT数组整体后移一位，这样就得到了next数组。原来next数组只是一个提供便利的“衍生物”。如下图所示：<br>
<img src="https://zy-zzf2000.github.io//post-images/1572086367096.jpg" alt="" loading="lazy"><br>
在上图中可以发现next【0】被设置为了-1，这是为了便于编程，使我们知道主串在第一位便发生了失配，这时候，i指针就要向后移动一位。</p>
<hr>
<h1 id="3-代码">3、代码</h1>
<p>既然明白了next数组的由来，我个人认为实现它也就不困难了。</p>
<pre><code class="language-c++">int KMP(char* t,char* p){
int i=0,j=0;;
while(i<strlen(t)&&j<strlen(p)){
if(j==-1){ //如果j=-1，说明主串第一个字符就与模式字符串失配
i++; //主串指针下移一位
j=0; //模式字符串指针回溯到0
}
else if(t[i]==p[j]){ //如果当前位相等，则继续比较下一位
i++;
j++;
}
else{ //发生失配
j=next[j]; //模式字符串回溯到next[j]处
}
}
}
</code></pre>
<p>当然，如何实现next数组甚至比KMP算法更关键<br>
这里我直接引用知乎上的说法：</p>
<blockquote>
<p>其实，求next数组的过程完全可以看成字符串匹配的过程，即以模式字符串为主字符串，以模式字符串的前缀为目标字符串，一旦字符串匹配成功，那么当前的next值就是匹配成功的字符串的长度。<br>
具体来说，就是从模式字符串的第一位(注意，不包括第0位)开始对自身进行匹配运算。 在任一位置，能匹配的最长长度就是当前位置的next值。</p>
</blockquote>
<p>我只补充一点，那就是匹配成功之后，模式字符串的指针j+1就是所谓最长长度，也就是next的值。</p>
<p>下面是几张图片：<br>
<img src="https://zy-zzf2000.github.io//post-images/1572100503477.jpg" alt="" loading="lazy"><br>
<img src="https://zy-zzf2000.github.io//post-images/1572100507615.jpg" alt="" loading="lazy"><br>
<img src="https://zy-zzf2000.github.io//post-images/1572100511040.jpg" alt="" loading="lazy"><br>
<img src="https://zy-zzf2000.github.io//post-images/1572100514912.jpg" alt="" loading="lazy"><br>
<img src="https://zy-zzf2000.github.io//post-images/1572100518437.jpg" alt="" loading="lazy"><br>
实际上，无论什么时候，都有next【0】=-1，next【1】=0.这是因为，next数组是由PMT数组右移一位得来，我们约定next【0】=-1，便于编程；而next【1】的值，实际上是主串第一个字符最大重复元素的长度，显然为0.<br>
值得注意的是，计算next数组时，要从第一位开始，即第0位不参与，由于next数组是由PMT数组右移一位得来，那么，第一位所得到的值实际上是next【2】的值。<br>
这两段也就解释了为什么上图是从next【2】开始计算的。</p>
<pre><code class="language-c++">void getNext(char *p,int *next){
next[0]=-1; //next[0]被约定为-1
next[1]=0; //next[1]一定是0
int i=1,j=0;
while(i < strlen(p)-1) {
if (j == -1 || p[i] == p[j]){
++i;
++j;
next[i] = j;
}
else{
j = next[j];
}
}
}
</code></pre>
]]></content>
</entry>
<entry>
<title type="html"><![CDATA[逆波兰算法实现简单优先计算器]]></title>
<id>https://zy-zzf2000.github.io/post/逆波兰</id>
<link href="https://zy-zzf2000.github.io/post/逆波兰">
</link>
<updated>2019-10-17T14:43:18.000Z</updated>
<content type="html"><![CDATA[<blockquote>
<p>1、只实现了+、-、*、/、（、）的优先级运算<br>
2、实现了多位数的运算</p>
</blockquote>
<pre><code class="language-c++">#include <iostream>
#include <iomanip>
#include <stack>
#include <vector>
#include <stdlib.h>
#include <sstream>
using namespace std;
const static char priority[7][7] = //丑陋的优先级比较方式
{
{ '=','=','<','<','>','0','>' },
{ '=','=','<','<','>','0','>' },
{ '>','>','=','=','>','0','>' },
{ '>','>','=','=','>','0','>' },
{ '<','<','<','<','<','=','0' },
{ '0','0','0','0','0','0','0' },
{ '<','<','<','<','<','0','=' },
};
int getIndex(string ch){ //将运算符转换成相应的下标
if(ch=="+"){
return 0;
}
if(ch=="-"){
return 1;
}
if(ch=="*"){
return 2;
}
if(ch=="/"){
return 3;
}
if(ch=="("){
return 4;
}
if(ch==")"){
return 5;
}
if(ch=="#"){
return 6;
}
}
bool isDegital(string str) { //判断一个string是否为整数
for (int i = 0;i < str.size();i++) {
if (str.at(i) == '-' && str.size() > 1) // 有可能出现负数
continue;
if (str.at(i) > '9' || str.at(i) < '0')
return false;
}
return true;
}
int to_int(string ch){ //将string转换为整数
stringstream ss;
int value;
ss<<ch;
ss>>value;
return value;
}
stack<string> switchIt(string str){ //将传输过来的表达式转换为逆序
stack<string> s1;
stack<string> s2;
stack<string> s3;
s1.push("#");
int counter = 0;
for(int i=0;i<str.length();i++){
string ch(1,str[i]);
if(isDegital(ch)){
if(counter==1){
string t=s2.top();
s2.pop();
int val1,val2,new_val;
stringstream ss1,ss2,ss3;
ss1<<t;
ss1>>val1;
ss2<<ch;
ss2>>val2;
new_val=val1*10+val2;
string new_str;
ss3<<new_val;
new_str=ss3.str();
s2.push(new_str);
}
else{
s2.push(ch);
counter++;
}
}
else if(ch=="("){
counter=0;
s1.push(ch);
}
else if(ch==")"){
counter=0;
while(s1.top()!="("){
s2.push(s1.top());
s1.pop();
}
s1.pop();
}
else{
counter=0;
int index1=getIndex(ch);
int index2=getIndex(s1.top());
char cmp=priority[index1][index2];
if(cmp=='>'){
s1.push(ch);
}
else{
while(cmp!='>'){
s2.push(s1.top());
s1.pop();
int index1=getIndex(ch);
int index2=getIndex(s1.top());
cmp=priority[index1][index2];
}
s1.push(ch);
}
}
}
while(!s1.empty()&&s1.top()!="#"){
s2.push(s1.top());
s1.pop();
}
while(!s2.empty()){
s3.push(s2.top());
s2.pop();
}
return s3;
}
int main(){
cout << "Please input an expression:" << endl;
string str;
cin>>str;
stack<string> res=switchIt(str);
stack<int> values;
// while(!res.empty()){
// cout<<res.top();
// res.pop();
// }
while(!res.empty()){
string ch=res.top();
int counter=0;
if(isDegital(ch)){
int value=to_int(ch);
values.push(value);
}
else if(ch=="+"){
int val1=values.top();
values.pop();
int val2=values.top();
values.pop();
values.push(val1+val2);
cout<<val2<<"+"<<val1<<"="<<val1+val2<<endl;
}
else if(ch=="-"){
int val1=values.top();
values.pop();
int val2=values.top();
values.pop();
values.push(val2-val1);
cout<<val2<<"-"<<val1<<"="<<val2-val1<<endl;
}
else if(ch=="*"){
int val1=values.top();
values.pop();
int val2=values.top();
values.pop();
values.push(val1*val2);
cout<<val2<<"*"<<val1<<"="<<val2*val1<<endl;
}
else if(ch=="/"){
int val1=values.top();
values.pop();
int val2=values.top();
values.pop();
if(val1==0){
cout<<"error:The divisor is 0.";
exit(0);
}
values.push(val2/val1);
cout<<val2<<"/"<<val1<<"="<<val2/val1<<endl;
}
res.pop();
}
cout<<"The result of the expression:"<<values.top();
return 0;
}
</code></pre>
]]></content>
</entry>
</feed>