fD1.m

function fd_1st_d = fD1(X, D, A, N, d)

% ---------------------------------------------------------------------------
% the gradient of the dissimilarity constraint function w.r.t. A
%
% for example, let distance by L1 norm:
% f = f(\sum_{ij \in D} \sqrt{(x_i-x_j)A(x_i-x_j)'})
% df/dA_{kl} = f'* d(\sum_{ij \in D} \sqrt{(x_i-x_j)^k*(x_i-x_j)^l})/dA_{kl}
%
% note that d_ij*A*d_ij' = tr(d_ij*A*d_ij') = tr(d_ij'*d_ij*A)
% so, d(d_ij*A*d_ij')/dA = d_ij'*d_ij
%     df/dA = f'(\sum_{ij \in D} \sqrt{tr(d_ij'*d_ij*A)})
%             * 0.5*(\sum_{ij \in D} (1/sqrt{tr(d_ij'*d_ij*A)})*(d_ij'*d_ij))
% ---------------------------------------------------------------------------
           
sum_dist = 0.000001; sum_deri =  zeros(d,d); 

for i = 1:N
  for j= i+1:N     % count each pair once
    if D(i,j) == 1
      d_ij = X(i,:) - X(j,:);
      [dist_ij, deri_d_ij] = distance1(A, d_ij);
      sum_dist = sum_dist +  dist_ij;
      sum_deri = sum_deri + deri_d_ij;
    end  
  end
end
%sum_dist
fd_1st_d = dgF2(sum_dist)*sum_deri;

% ------------------------------------------------


% ___________derivative of cover function 1_________
function z = dgF1(y)
z = 1;

% ___________derivative of cover function 2_________
function z = dgF2(y)
z = 1/y;



function [dist_ij, deri_d_ij] = distance1(A, d_ij)
% distance and derivative of distance using distance1: distance(d) = L1
fudge = 0.000001;  % regularizes derivates a little

      M_ij = d_ij'*d_ij;
      dist_ij = sqrt(trace(M_ij*A));

      % derivative of dist_ij w.r.t. A
      deri_d_ij = 0.5*M_ij/(dist_ij+fudge); 


function [dist_ij, deri_d_ij] = distance2(A, d_ij)
% distance and derivative of distance using distance2: distance(d) = sqrt(L1)
fudge = 0.000001;  % regularizes derivates a little

      M_ij = d_ij'*d_ij;
      L2 = trace(M_ij*A);           % L2 norm
      dist_ij = sqrt(sqrt(L2));

      % derivative of dist_ij w.r.t. A
      deri_d_ij = 0.25*M_ij/(L2^(3/4)+fudge); 


function [dist_ij, deri_d_ij] = distance3(A, d_ij)
% distance and derivative of distance using distance3: 1-exp(-\beta*L1)
fudge = 0.000001;  % regularizes derivates a little

      beta = 0.5;
      M_ij = d_ij'*d_ij;
      L1 = sqrt(trace(M_ij*A));
      dist_ij = 1 - exp(-beta*L1);

      % derivative of dist_ij w.r.t. A
      deri_d_ij = 0.5*beta*exp(-beta*L1)*M_ij/(L1+fudge);