Plot the figure of K-SVCR

 

 

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clear   %% generate data    prettySpiral = 0;    if ~prettySpiral     % generate some random gaussian like data     rand('state', 0);     randn('state', 0);     N= 50;     D= 2;        X1 = mgd(N, D, [4 3], [2 -1;-1 2]);     X2 = mgd(N, D, [1 1], [2 1;1 1]);     X3 = mgd(N, D, [3 -3], [1 0;0 4]);        X= [X1; X2; X3];     X= bsxfun(@rdivide, bsxfun(@minus, X, mean(X)), var(X));     y= [ones(N, 1); ones(N, 1)*2; ones(N, 1)*3];        scatter(X(:,1), X(:,2), 20, y)    else     % generate twirl data!        N= 50;     t = linspace(0.5, 2*pi, N);     x = t.*cos(t);     y = t.*sin(t);        t = linspace(0.5, 2*pi, N);     x2 = t.*cos(t+2);     y2 = t.*sin(t+2);        t = linspace(0.5, 2*pi, N);     x3 = t.*cos(t+4);     y3 = t.*sin(t+4);        X= [[x' y']; [x2' y2']; [x3' y3']];     X= bsxfun(@rdivide, bsxfun(@minus, X, mean(X)), var(X));     y= [ones(N, 1); ones(N, 1)*2; ones(N, 1)*3];        scatter(X(:,1), X(:,2), 20, y) end   %% classify rho = 1; c1 =10; c2 =10; epsilon = 0.2; threshold = 0;   result=[]; %  ker = 'linear';  ker = 'rbf'; sigma = 1/10;%sigma = 1/200; par = NonLinearDualBoundSVORIM(X, y, c1, c2, epsilon, rho, ker, sigma); %f = TestPrecisionNonLinear(par,X, y,X, y, ker,epsilon,sigma);   %% Plot the figure contour_level = [-epsilon,0, epsilon]; xrange = [-1.5 1.5]; yrange = [-1.5 1.5]; % step size for how finely you want to visualize the decision boundary. inc = 0.005; % generate grid coordinates. this will be the basis of the decision % boundary visualization. [x1, x2] = meshgrid(xrange(1):inc:xrange(2), yrange(1):inc:yrange(2)); % size of the (x, y) image, which will also be the size of the % decision boundary image that is used as the plot background. image_size = size(x1);   xy = [x1(:) x2(:)]; % make (x,y) pairs as a bunch of row vectors.   % set up the domain over which you want to visualize the decision % boundary % d = []; % for k=1:max(y) %     par.normw{k}=1; %     d(:,k) =  decisionfun(xy, par, X,y,k,epsilon, ker,sigma)'; % end % [~,idx] = min(abs(d)/par.normw{k},[],2); % nd=max(y);   nd = (max(y)*(max(y)-1)/2); d = []; pred=zeros(size(xy,1),nd); for k=1:nd     par.normw{k}=1;     d(:,k) =  decisionfun(xy, par, X,y,k,epsilon, ker,sigma)'; end pred(d<-threshold) = -1; pred(d >threshold) = 1; nclass = max(y); expLosses=zeros(size(pred,1),nclass);   for i=1:nclass,     expLosses(:,i) = sum(pred == repmat(par.Code(i,:),size(pred,1),1),2); end [minVal,finalOutput] = max(expLosses,[],2); idx = finalOutput;   plt = 2; %1, just show the decison region with different colors; 2, show the decision hyperlane between class 1 and class 3 switch plt     case 1         % reshape the idx (which contains the class label) into an image.         decisionmap = reshape(idx, image_size);          imagesc(xrange,yrange,decisionmap);         % plot the class training data.         hold on;         set(gca,'ydir','normal');         cmap = [1 0.8 0.8; 0.95 1 0.95; 0.9 0.9 1];         colormap(cmap);         plot(X(y==1,1), X(y==1,2), 'o', 'MarkerFaceColor', [.9 .3 .3], 'MarkerEdgeColor','k');         plot(X(y==2,1), X(y==2,2), 'o', 'MarkerFaceColor', [.3 .9 .3], 'MarkerEdgeColor','k');         plot(X(y==3,1), X(y==3,2), 'o', 'MarkerFaceColor', [.3 .3 .9], 'MarkerEdgeColor','k');         hold on;         % title(sprintf('%d trees, Train time: %.2fs, Test time: %.2fs\n', opts.numTrees, timetrain, timetest));     case 2         %% show SVs         color = {[.9 .3 .3],[.3 .9 .3],[.3 .3 .9]};         SVs = (par.SVs{2}>1e-4);         for i=1:max(y)             % show the SVs using biger marker             plot(X(y==i&SVs==1,1),X(y==i&SVs==1,2), 'o', 'MarkerFaceColor', color{i}, 'MarkerEdgeColor','k');             hold on             % plot the points of not SVs             plot(X(y==i&SVs~=1,1),X(y==i&SVs~=1,2), 'o', 'MarkerFaceColor', color{i}, 'MarkerEdgeColor',color{i});          end         hold on;         title(sprintf('Ratio of SVs is %.2fs\n', mean(SVs)));         color = {'r-','g-','b*','r.','go','b*'};         color1 = {'r-','g--','b*','r.','go','b*'};         contour_level1 = [-epsilon, 0, epsilon];         contour_level2 = [-epsilon, 0, epsilon];         contour_level0 = [-1,0,1];         % for k = 1:nd         for k=2             decisionmapk = reshape(d(:,k), image_size);             contour(x1,x2, decisionmapk, [-1 1], color1{k},'LineWidth',0.5);             contour(x1,x2, decisionmapk, [contour_level(1) contour_level(1)], color{k});             contour(x1,x2, decisionmapk, [contour_level(2) contour_level(2) ], color{k},'LineWidth',2);             contour(x1,x2, decisionmapk, [contour_level(3) contour_level(3) ], color{k});             contour(x1,x2, decisionmapk, [contour_level0(3) contour_level0(3) ], color1{k},'LineWidth',0.5);         end end

 

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function par = NonLinearDualBoundSVORIM(traindata, targets, c1, c2, epsilon, rho, ker, sigma) %'traindata' is a training data matrix , each line is a sample vector %'targets' is a label vector,should  start  from 1 to p model='EX'; % rho is the augmented Lagrangian parameter.   % history is a structure that contains the objective value, the primal and % dual residual norms, and the tolerances for the primal and dual residual % norms at each iteration.   %Data preprocessing [n, m] = size(traindata); Lab=sort(unique(targets)); p=length(Lab); %  the number of total rank l= zeros(1,p); id={}; X=[];Y=[]; i=1; Id = []; while i<=p     id{i}=find(targets==Lab(i));     l(i)=length(id{i});     X=[X;traindata(id{i},:)];     Y=[Y;targets(id{i})];     Id = [Id, id{i}];     i=i+1; end [~,Id0]=sort(Id);    lc=cumsum(l);     w = [];  b = [];     s = cell(1,p);  r = cell(1,p);    K=Kernel(ker, X',X',sigma); K=(K+K')/2;   nch =  nchoosek([1:p],2); Code = zeros(p,size(nch,1));   for k =1:size(nch,1)     Code(nch(k,:),k) = [-1,1];     i = nch(k,1); j =nch(k,2);     s{k} =lc(i)-l(i)+1:lc(i);     r{k} = lc(j)-l(j)+1:lc(j);     c{k} = [1:n];     c{k}([lc(i)-l(i)+1:lc(i)  lc(j)-l(j)+1:lc(j)]) = [];         Ak = X(c{k},:);         Lk = X(s{k},:);         Rk =X(r{k},:);         row=[c{k} c{k} s{k} r{k}];         Hk = K(row,row);         %    model= subSVOR(Ak,Hk,Lk,Rk,c1, c2, epsilon, rho);         model = subSVOR_quadgrog(Ak, Hk, Lk,Rk,c1, c2, epsilon);                   P{k} = model.P;         p0{k} = model.p;         alpha{k} = model.alpha;         alphax{k}= model.alphax;         normw{k} = model.normw;         time(k) = model.time;         b(k,1) = model.b;         Id1= [s{k} c{k} r{k}];         SVs{k}= model.SVs(Id1); end    par.l= s;    par.r = r;    par.c= c;    par.P = P;    par.p = p0;    par.alpha = alpha;    par.alphax = alphax;    par.normw = normw;    par.time = time;    par.b = b;    par.X=X;    par.maxtime = max(par.time);     par.SVs =  SVs;     par.Y=Y;     par.Code = Code; %    par.w = w; %    par.b = b;   end     function par = subSVOR_quadgrog(Ak, H, Lk,Rk, c1, c2, epsilon)   t_start = tic; %Global constants and defaults QUIET    = 0;   m = size(Ak,2); lk = size(Ak,1); rk1 = size(Lk,1); rk2 =  size(Rk,1); rk = rk1+rk2; %ADMM solver mP=2*lk +rk;    %dimension of Phi mG=4*lk + 2*rk;  %dimension of Gamma mU=mG+1;                                %dimension of U mp1 = 1 : lk;        mp2 = lk+1 : 2*lk; mp3 = 2*lk+1: mP;   c= zeros(mU,1); c([mp1 mp2]+1) = c1; c(mp3+1) = c2;   q = ones(mP,1); q(mp1) = epsilon; q(mp2) = epsilon; q(mp3) = -1;   p = ones(mP,1); p(mp2) = -1; p(mP-rk2+1:mP) =  -1;   % H = [Ak; -Ak; Bk{1}; -Bk{2}]*[Ak; -Ak; Bk{1}; -Bk{2}]';   %linear Kernel % Qk=[Ak; Ak; Lk; Rk];   % H= Kernel(ker, Qk',Qk',sigma); % H=(H'+H)/2+1; H0 = (H+1).*(p*p');   % % options = optimoptions('quadprog',... % %     'Algorithm','interior-point-convex','Display','off'); options = optimoptions('quadprog',...     'Algorithm','interior-point-convex','MaxIter',200,'Display','off'); % % x = quadprog(H,f,A,b,Aeq,beq,lb,ub,x0,options) A = []; b = []; f = q; Aeq =[]; beq = []; lb = zeros(mP,1); ub = c(2:mP+1); x0 = [];  P = quadprog(H0,f,A,b,Aeq,beq,lb,ub,x0,options);    % diagnostics, reporting, termination checks     par.P= P;  par.p= p;  par.alpha = P(mp1);  par.alphax = P(mp2);  P3=P(mp3);  par.SVs = [P3(1:rk1);abs(P(mp1)-P(mp2));P3(rk1+1:rk1+rk2)]; %  par.normw = sqrt(P'*(H0.*(p'*p))*P);   par.normw =1;  bk =(p'*P);  par.b =bk;   %  switch ker %      case 'linear' %   par.w = [Ak; -Ak; Bk{1}; -Bk{2}]'*P; %   b1 = Ak(P(mp1)~=0,:)* par.w-epsilon;   %   b2 = Ak(P(mp2)~=0,:)* par.w+epsilon; %   par.b = mean([b1;b2]); %  end       if  ~QUIET        par.time = toc(t_start);  end   end   function [par, history]  = subSVOR(Ak,H, Lk,Rk, c1, c2, epsilon, rho) %'traindata' is a training data matrix , each line is a sample vector %'targets' is a label vector,should  start  from 1 to p   % rho is the augmented Lagrangian parameter.   % history is a structure that contains the objective value, the primal and % dual residual norms, and the tolerances for the primal and dual residual % norms at each iteration.   t_start = tic;   %Data preprocessing   %Global constants and defaults QUIET    = 0; MAX_ITER = 200; % ABSTOL   = 1e-4; % RELTOL   = 1e-2; ABSTOL   = 1e-6; RELTOL   = 1e-3;   lk = size(Ak,1); rk1 = size(Lk,1); rk2 =  size(Rk,1); rk = rk1+rk2;   %ADMM solver mP=2*lk +rk;    %dimension of Phi mG=4*lk + 2*rk;  %dimension of Gamma mU=mG;                                %dimension of U P = zeros(mP,1);                       %Phi={ w,b, xi, xi*} G = zeros(mG,1);                       %Gamma={eta,eta*,delta, phi, phi*} U = zeros(mU,1);                       %U- -update   mp1 = 1 : lk;        mp2 = lk+1 : 2*lk; mp3 = 2*lk+1: mP;   c= zeros(mU,1); c([mp1 mp2]) = c1; c(mp3) = c2;   q = ones(mP,1); q(mp1) = epsilon; q(mp2) = epsilon; q(mp3) = -1;   p = ones(mP,1); p(mp2) = -1; p(mP-rk2+1:mP) =  -1;   % H = [Ak; -Ak; Bk{1}; -Bk{2}]*[Ak; -Ak; Bk{1}; -Bk{2}]';   %linear Kernel   %  Qk=[Ak; Ak; Lk; Rk];   % H0= Kernel(ker, Qk',Qk',sigma);%Kernel Matrix of Bound SVM % H = (H0+1).*(p*p'); % H0= Kernel(ker, Qk',Qk',sigma);%Kernel Matrix of Bound SVM H = (H+1).*(p*p');   k=1; while k <=MAX_ITER     %Phi={ w,b, xi, xi*}-update     V = U + B(mP,G) - c;     br = - q - rho * AtX(mP,V);     [P, niters] = cgsolve(H, br, rho);           %Gamma={eta,eta*,delta, phi, phi*}-update with relaxation     Gold = G;     G = pos(Bt(mP,c-AX(P)-U));           %U- -update     r = AX(P) + B(mP,G) - c;     U = U + r;     %     history.objval(k)  = objective(H,P,q);     s = rho*AtX(mP,B(mP,G - Gold));     history.r_norm(k) = norm(r);     history.s_norm(k) = norm(s);       history.eps_pri(k) = sqrt(mU)*ABSTOL + RELTOL*max([norm(AX(P)), norm(B(mP,G)), norm(c)]);     history.eps_dual(k)= sqrt(mP)*ABSTOL + RELTOL*norm(rho*AtX(mP,U));           if  history.r_norm(k) < history.eps_pri(k) && ...          history.s_norm(k) < history.eps_dual(k);          break     end     k = k+1; end     if  ~QUIET        par.time = toc(t_start);  end     par.P= P;  par.p= p;  par.alpha = P(mp1);  par.alphax = P(mp2); % par.normw = sqrt(P'*(H0.*(p'*p))*P); par.normw=1; bk =(p'*P);   par.b =bk;     if  ~QUIET        par.time = toc(t_start);  end    end   % function obj = objective(H,P,q) %     obj = 1/2 * vHv(H,P) + q'*P; % end   function [x, niters] = cgsolve(H, b,rho,tol, maxiters) % cgsolve : Solve Ax=b by conjugate gradients % % Given symmetric positive definite sparse matrix A and vector b, % this runs conjugate gradient to solve for x in A*x=b. % It iterates until the residual norm is reduced by 10^-6, % or for at most max(100,sqrt(n)) iterations   n = length(b);   if (nargin < 4)     tol = 1e-6;     maxiters = max(100,sqrt(n)); elseif(nargin < 5)     maxiters = max(100,sqrt(n)); end   normb = norm(b); x = zeros(n,1); r = b; rtr = r'*r; d = r; niters = 0; while sqrt(rtr)/normb > tol  &&  niters < maxiters     niters = niters+1; %     Ad = A*d;     Ad = AtAX(H, d,rho);     alpha = rtr / (d'*Ad);     x = x + alpha * d;     r = r - alpha * Ad;     rtrold = rtr;     rtr = r'*r;     beta = rtr / rtrold;     d = r + beta * d; end end   %Ad = A*d function Ad = AtAX(H, d,rho) Ad = H*d +2* rho*d; end   function F = AtX(mP,V) F = V(1:mP)+V(mP+1:end); end   function h = AX(P) h = [P;P]; end     function h = vHv(H,d) h = d'*(H*d) ; end   function Bv= B(mP,v) Bv(:,1) = [v(1:mP);-v(mP+1:end)]; end   function Btd = Bt(mP,d) Btd = [d(1:mP);-d(mP+1:end)]; end   function A = pos(A) A(A<0)=0; end