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
时间： 2024-10-27 10:55:24