softth

 SOFTTH - Computes the proximity operator with respect to the
          trace norm

 Syntax
  function [vv,ss,nsv]=softth(vv, lambda, nsv, verbose);

 Reference
 "On the extension of trace norm to tensors"
 Ryota Tomioka, Kohei Hayashi, and Hisashi Kashima
 arXiv:1010.0789
 http://arxiv.org/abs/1010.0789
 
 Copyright(c) 2010 Ryota Tomioka
 This software is distributed under the MIT license. See license.txt

0001 % SOFTTH - Computes the proximity operator with respect to the
0002 %          trace norm
0003 %
0004 % Syntax
0005 %  function [vv,ss,nsv]=softth(vv, lambda, nsv, verbose);
0006 %
0007 % Reference
0008 % "On the extension of trace norm to tensors"
0009 % Ryota Tomioka, Kohei Hayashi, and Hisashi Kashima
0010 % arXiv:1010.0789
0011 % http://arxiv.org/abs/1010.0789
0012 %
0013 % Copyright(c) 2010 Ryota Tomioka
0014 % This software is distributed under the MIT license. See license.txt
0015 
0016 
0017 function [vv,ss,nsv]=softth(vv, lambda, nsv, verbose);
0018 
0019 if ~exist('verbose','var')
0020   verbose=0;
0021 end
0022 
0023 
0024 sz=size(vv);
0025 nsv=min(min(sz),nsv+1);
0026 
0027 if verbose
0028   fprintf('sz=[%d %d]\n',sz(1), sz(2));
0029   fprintf('nsv=');
0030 end
0031   
0032 while 1
0033   if verbose
0034     fprintf('%d/',nsv);
0035   end
0036  [U,S,V]=pca(vv,min(min(sz),nsv),10);
0037  ss=diag(S);
0038   if min(ss)<lambda || nsv==min(sz)
0039     if verbose
0040       fprintf('min(ss)=%g\n',min(ss));
0041     end
0042     break;
0043   else
0044     nsv=min(min(sz),round(nsv*2));
0045   end
0046 end
0047 
0048 ix=find(ss>=lambda);
0049 ss=ss(ix)-lambda;
0050 vv = U(:,ix)*diag(ss)*V(:,ix)';
0051 
0052 nsv=length(ix);