AP(affinity propagation）研究
待补充……
AP算法，即Affinity propagation，是Brendan J. Frey* 和Delbert Dueck于2007年在science上提出的一种算法（文章链接，维基百科）
现在只是初步研究了一下官网上提供的MATLAB源码：apcluster.m
%APCLUSTER Affinity Propagation Clustering (Frey/Dueck, Science 2007) % [idx,netsim,dpsim,expref]=APCLUSTER(s,p) clusters data, using a set % of real-valued pairwise data point similarities as input. Clusters % are each represented by a cluster center data point (the "exemplar"). % The method is iterative and searches for clusters so as to maximize % an objective function, called net similarity. % % For N data points, there are potentially N^2-N pairwise similarities; % this can be input as an N-by-N matrix 's', where s(i,k) is the % similarity of point i to point k (s(i,k) needn抰 equal s(k,i)). In % fact, only a smaller number of relevant similarities are needed; if % only M similarity values are known (M < N^2-N) they can be input as % an M-by-3 matrix with each row being an (i,j,s(i,j)) triple. % % APCLUSTER automatically determines the number of clusters based on % the input preference 'p', a real-valued N-vector. p(i) indicates the % preference that data point i be chosen as an exemplar. Often a good % choice is to set all preferences to median(s); the number of clusters % identified can be adjusted by changing this value accordingly. If 'p' % is a scalar, APCLUSTER assumes all preferences are that shared value. % % The clustering solution is returned in idx. idx(j) is the index of % the exemplar for data point j; idx(j)==j indicates data point j % is itself an exemplar. The sum of the similarities of the data points to % their exemplars is returned as dpsim, the sum of the preferences of % the identified exemplars is returned in expref and the net similarity % objective function returned is their sum, i.e. netsim=dpsim+expref. % % [ ... ]=apcluster(s,p,'NAME',VALUE,...) allows you to specify % optional parameter name/value pairs as follows: % % 'maxits' maximum number of iterations (default: 1000) % 'convits' if the estimated exemplars stay fixed for convits % iterations, APCLUSTER terminates early (default: 100) % 'dampfact' update equation damping level in [0.5, 1). Higher % values correspond to heavy damping, which may be needed % if oscillations occur. (default: 0.9) % 'plot' (no value needed) Plots netsim after each iteration % 'details' (no value needed) Outputs iteration-by-iteration % details (greater memory requirements) % 'nonoise' (no value needed) APCLUSTER adds a small amount of % noise to 's' to prevent degenerate cases; this disables that. % % Copyright (c) B.J. Frey & D. Dueck (2006). This software may be % freely used and distributed for non-commercial purposes. % (RUN APCLUSTER WITHOUT ARGUMENTS FOR DEMO CODE) function [idx,netsim,dpsim,expref]=apcluster(s,p,varargin); if nargin==0, % display demo fprintf('Affinity Propagation (APCLUSTER) sample/demo code\n\n'); fprintf('N=100; x=rand(N,2); % Create N, 2-D data points\n'); fprintf('M=N*N-N; s=zeros(M,3); % Make ALL N^2-N similarities\n'); fprintf('j=1;\n'); fprintf('for i=1:N\n'); fprintf(' for k=[1:i-1,i+1:N]\n'); fprintf(' s(j,1)=i; s(j,2)=k; s(j,3)=-sum((x(i,:)-x(k,:)).^2);\n'); fprintf(' j=j+1;\n'); fprintf(' end;\n'); fprintf('end;\n'); fprintf('p=median(s(:,3)); % Set preference to median similarity\n'); fprintf('[idx,netsim,dpsim,expref]=apcluster(s,p,''plot'');\n'); fprintf('fprintf(''Number of clusters: %%d\\n'',length(unique(idx)));\n'); fprintf('fprintf(''Fitness (net similarity): %%g\\n'',netsim);\n'); fprintf('figure; % Make a figures showing the data and the clusters\n'); fprintf('for i=unique(idx)''\n'); fprintf(' ii=find(idx==i); h=plot(x(ii,1),x(ii,2),''o''); hold on;\n'); fprintf(' col=rand(1,3); set(h,''Color'',col,''MarkerFaceColor'',col);\n'); fprintf(' xi1=x(i,1)*ones(size(ii)); xi2=x(i,2)*ones(size(ii)); \n'); fprintf(' line([x(ii,1),xi1]'',[x(ii,2),xi2]'',''Color'',col);\n'); fprintf('end;\n'); fprintf('axis equal tight;\n\n'); return; end; start = clock; % Handle arguments to function if nargin<2 error('Too few input arguments'); else maxits=1000; convits=100; lam=0.9; plt=0; details=0; nonoise=0; i=1; while i<=length(varargin) if strcmp(varargin{i},'plot') plt=1; i=i+1; elseif strcmp(varargin{i},'details') details=1; i=i+1; elseif strcmp(varargin{i},'sparse') % [idx,netsim,dpsim,expref]=apcluster_sparse(s,p,varargin{:}); fprintf('''sparse'' argument no longer supported; see website for additional software\n\n'); return; elseif strcmp(varargin{i},'nonoise') nonoise=1; i=i+1; elseif strcmp(varargin{i},'maxits') maxits=varargin{i+1}; i=i+2; if maxits<=0 error('maxits must be a positive integer'); end; elseif strcmp(varargin{i},'convits') convits=varargin{i+1}; i=i+2; if convits<=0 error('convits must be a positive integer'); end; elseif strcmp(varargin{i},'dampfact') lam=varargin{i+1}; i=i+2; if (lam<0.5)||(lam>=1) error('dampfact must be >= 0.5 and < 1'); end; else i=i+1; end; end; end; if lam>0.9 fprintf('\n*** Warning: Large damping factor in use. Turn on plotting\n'); fprintf(' to monitor the net similarity. The algorithm will\n'); fprintf(' change decisions slowly, so consider using a larger value\n'); fprintf(' of convits.\n\n'); end; % Check that standard arguments are consistent in size if length(size(s))~=2 error('s should be a 2D matrix'); elseif length(size(p))>2 error('p should be a vector or a scalar'); elseif size(s,2)==3 tmp=max(max(s(:,1)),max(s(:,2))); if length(p)==1 N=tmp; else N=length(p); end; if tmp>N error('data point index exceeds number of data points'); elseif min(min(s(:,1)),min(s(:,2)))<=0 error('data point indices must be >= 1'); end; elseif size(s,1)==size(s,2) N=size(s,1); if (length(p)~=N)&&(length(p)~=1) error('p should be scalar or a vector of size N'); end; else error('s must have 3 columns or be square'); end; % Construct similarity matrix if N>3000 fprintf('\n*** Warning: Large memory request. Consider activating\n'); fprintf(' the sparse version of APCLUSTER.\n\n'); end; if size(s,2)==3 && size(s,1)~=3, S=-Inf*ones(N,N,class(s)); for j=1:size(s,1), S(s(j,1),s(j,2))=s(j,3); end; else S=s; end; if S==S', symmetric=true; else symmetric=false; end; realmin_=realmin(class(s)); realmax_=realmax(class(s)); % In case user did not remove degeneracies from the input similarities, % avoid degenerate solutions by adding a small amount of noise to the % input similarities if ~nonoise rns=randn('state'); randn('state',0); S=S+(eps*S+realmin_*100).*rand(N,N); randn('state',rns); end; % Place preferences on the diagonal of S if length(p)==1 for i=1:N S(i,i)=p; end; else for i=1:N S(i,i)=p(i); end; end; % Numerical stability -- replace -INF with -realmax n=find(S<-realmax_); if ~isempty(n), warning('-INF similarities detected; changing to -REALMAX to ensure numerical stability'); S(n)=-realmax_; end; clear('n'); if ~isempty(find(S>realmax_,1)), error('+INF similarities detected; change to a large positive value (but smaller than +REALMAX)'); end; % Allocate space for messages, etc dS=diag(S); A=zeros(N,N,class(s)); R=zeros(N,N,class(s)); t=1; if plt, netsim=zeros(1,maxits+1); end; if details idx=zeros(N,maxits+1); netsim=zeros(1,maxits+1); dpsim=zeros(1,maxits+1); expref=zeros(1,maxits+1); end; % Execute parallel affinity propagation updates e=zeros(N,convits); dn=0; i=0; if symmetric, ST=S; else ST=S'; end; % saves memory if it's symmetric while ~dn i=i+1; % Compute responsibilities A=A'; R=R'; for ii=1:N, old = R(:,ii); AS = A(:,ii) + ST(:,ii); [Y,I]=max(AS); AS(I)=-Inf; [Y2,I2]=max(AS); R(:,ii)=ST(:,ii)-Y; R(I,ii)=ST(I,ii)-Y2; R(:,ii)=(1-lam)*R(:,ii)+lam*old; % Damping R(R(:,ii)>realmax_,ii)=realmax_; end; A=A'; R=R'; % Compute availabilities for jj=1:N, old = A(:,jj); Rp = max(R(:,jj),0); Rp(jj)=R(jj,jj); A(:,jj) = sum(Rp)-Rp; dA = A(jj,jj); A(:,jj) = min(A(:,jj),0); A(jj,jj) = dA; A(:,jj) = (1-lam)*A(:,jj) + lam*old; % Damping end; % Check for convergence E=((diag(A)+diag(R))>0); e(:,mod(i-1,convits)+1)=E; K=sum(E); if i>=convits || i>=maxits, se=sum(e,2); unconverged=(sum((se==convits)+(se==0))~=N); if (~unconverged&&(K>0))||(i==maxits) dn=1; end; end; % Handle plotting and storage of details, if requested if plt||details if K==0 tmpnetsim=nan; tmpdpsim=nan; tmpexpref=nan; tmpidx=nan; else I=find(E); notI=find(~E); [tmp c]=max(S(:,I),[],2); c(I)=1:K; tmpidx=I(c); tmpdpsim=sum(S(sub2ind([N N],notI,tmpidx(notI)))); tmpexpref=sum(dS(I)); tmpnetsim=tmpdpsim+tmpexpref; end; end; if details netsim(i)=tmpnetsim; dpsim(i)=tmpdpsim; expref(i)=tmpexpref; idx(:,i)=tmpidx; end; if plt, netsim(i)=tmpnetsim; figure(234); plot(((netsim(1:i)/10)*100)/10,'r-'); xlim([0 i]); % plot barely-finite stuff as infinite xlabel('# Iterations'); ylabel('Fitness (net similarity) of quantized intermediate solution'); % drawnow; end; end; % iterations I=find((diag(A)+diag(R))>0); K=length(I); % Identify exemplars if K>0 [tmp c]=max(S(:,I),[],2); c(I)=1:K; % Identify clusters % Refine the final set of exemplars and clusters and return results for k=1:K ii=find(c==k); [y j]=max(sum(S(ii,ii),1)); I(k)=ii(j(1)); end; notI=reshape(setdiff(1:N,I),[],1); [tmp c]=max(S(:,I),[],2); c(I)=1:K; tmpidx=I(c); tmpdpsim=sum(S(sub2ind([N N],notI,tmpidx(notI)))); tmpexpref=sum(dS(I)); tmpnetsim=tmpdpsim+tmpexpref; else tmpidx=nan*ones(N,1); tmpnetsim=nan; tmpexpref=nan; end; if details netsim(i+1)=tmpnetsim; netsim=netsim(1:i+1); dpsim(i+1)=tmpdpsim; dpsim=dpsim(1:i+1); expref(i+1)=tmpexpref; expref=expref(1:i+1); idx(:,i+1)=tmpidx; idx=idx(:,1:i+1); else netsim=tmpnetsim; dpsim=tmpdpsim; expref=tmpexpref; idx=tmpidx; end; if plt||details fprintf('\nNumber of exemplars identified: %d (for %d data points)\n',K,N); fprintf('Net similarity: %g\n',tmpnetsim); fprintf(' Similarities of data points to exemplars: %g\n',dpsim(end)); fprintf(' Preferences of selected exemplars: %g\n',tmpexpref); fprintf('Number of iterations: %d\n\n',i); fprintf('Elapsed time: %g sec\n',etime(clock,start)); end; if unconverged fprintf('\n*** Warning: Algorithm did not converge. Activate plotting\n'); fprintf(' so that you can monitor the net similarity. Consider\n'); fprintf(' increasing maxits and convits, and, if oscillations occur\n'); fprintf(' also increasing dampfact.\n\n'); end;
实际使用的示例数据：
s矩阵以及p的取值，
s=[1 0.85 0.9 0.5 0.45 0.5 0.4 0.4 0.5 0.45; 0.85 1 0.85 0.6 0.65 0.7 0.6 0.55 0.8 0.7; 0.9 0.85 1 0.75 0.7 0.65 0.55 0.5 0.6 0.5; 0.5 0.6 0.75 1 0.9 0.7 0.7 0.85 0.5 0.45; 0.45 0.65 0.7 0.9 1 0.9 0.9 0.85 0.6 0.65; 0.5 0.7 0.65 0.7 0.9 1 0.85 0.75 0.75 0.75; 0.4 0.6 0.55 0.7 0.9 0.85 1 0.85 0.5 0.55; 0.4 0.55 0.5 0.85 0.85 0.75 0.85 1 0.3 0.25; 0.5 0.8 0.6 0.5 0.6 0.75 0.5 0.3 1 0.9; 0.45 0.7 0.5 0.45 0.65 0.75 0.55 0.25 0.9 1; ]; p=median(median(s));
最后的运行结果：
idx =
1
1
1
5
5
5
5
5
9
9
netsim =
8.1875
dpsim =
6.2000
expref =
1.9875
