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NMFL0_H.m
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NMFL0_H.m
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function [W,H,INFO] = NMFL0_H(X, options)
%
% [W,H,INFO] = NMFL0_H(X, options)
%
% run NMFL0_H described in
% R. Peharz and F. Pernkopf, "Sparse nonnegative matrix factorization with
% ℓ0-constraints", Neurocomputing, 2012.
%
% The algorithm returns an approximate solution for
%
% minimize ||X - W * H||_F
% s.t. W(:) >= 0
% H(:) >= 0
% sum(H(:,k) > 0) <= L for all k
%
% w.r.t. W and H
%
%
% input:
%
% X nonnegative data matrix
% options: structure of parameters:
% K: number of columns in dictionary matrix W
% L: maximal number of nonzeros in each column of H
% numIter: number of (outer) iterations
% sparseCoder: function handle to sparse coder; sparse coder function
% must have the form H = foo(X,W,L,[params]), and
% return an approximation to the problem
%
% minimize ||X - W * H||_F
% s.t. W(:) >= 0
% H(:) >= 0
% sum(H(:,k) > 0) <= L for all k
%
% w.r.t. H.
% sparseCoderParams: params to be passed to the sparse coder. If not
% defined, the sparse coder will be called without params
% updateType: update type for dictionary W updates.
% - 'MU'multiplicative updates according to Lee and Seung,
% "Algorithms for nonnegative matrix factorization", 2001.
% - 'ANLS_FC': alternating nonnegative least squares, using
% fast combinatorial approach for NNLS, by
% M . H. Van Benthem and M. R. Keenan, "Fast algorithm
% for the solution of large-scale
% non-negativity-constrained least squares problems",
% Journal of Chemometrics, 2004
% - 'ANLS_PG': alternating nonnegative least squares, using
% projected gradient approach by Chih-Jen Lin,
% "Projected Gradient Methods for Nonnegative Matrix
% Factorization", Neural Computation, 2007.
% For this update method, options must also contain the
% fields
% NNLS_PG_tolerance
% NNLS_PG_maxIter
% (see NNLS_PG.m).
% - 'NNKSVD': dictionary updates from Aharon and Bruckstein,
% "K-SVD and its non-negative variant for dictionary
% design", SPIE, 2005.
% numUpdateIter: number of update (innner) iterations; only relevant if
% timeBudgetUpdate is empty
% timeBudgetUpdate: if not empty, this contains a vector of length
% numIter, containing the time budget (in seconds)
% to be used for the dictionary update in each outer
% iteration.
% VERBOSITY: verbose mode if not 0; default 1.
% W: initial dictionary to be used ([] if none).
% initType: dictionary initialisation if field W is empty;
% 'rand' for random numbers or 'samples' for data
% samples (default)
%
%
% output:
%
% W: dictionary matrix
% H: coding matrix
% INFO: structure of some info
% E_SC: error after sparse coder in each iteration (||X - W*H||_F)
% E: error in each iteration ||X - W*H||_F)
% SCtime: time needed by sparse coder
% UDtime: time needed by dictionary updates
%
% Robert Peharz, 2011
%
if any(X(:)<0), error('X contains negative values.'); end
if ~isfield(options,'K'), error('options must contain parameter K (number of basis vectors).'); end
if ~isfield(options,'L'), error('options must contain parameter L (maximal number of basis vectors per data sample).'); end
if ~isfield(options,'numIter'), error('options must contain parameter numIter.'); end
if ~isfield(options,'sparseCoder'), error('options must contain parameter sparseCoder (function_handle).'); end
if ~isfield(options,'updateType'), error('options must contain parameter updateType (string).'); end
if ~isfield(options,'numUpdateIter') && ~isfield(options,'timeBudgetUpdate')
error('options must contain either parameter numUpdateIter or timeBudgetUpdate.');
end
switch options.updateType
case 'MU'
case 'ANLS_FC'
case 'ANLS_PG'
if ~isfield(options,'NNLS_PG_tolerance'), error('options must contain parameter NNLS_PG_tolerance for updateType == ''ANLS_PG''.'); end
if ~isfield(options,'NNLS_PG_maxIter'), error('options must contain parameter NNLS_PG_maxIter for updateType == ''ANLS_PG''.'); end
NNLS_PG_tolerance = options.NNLS_PG_tolerance;
NNLS_PG_maxIter = options.NNLS_PG_maxIter;
case 'NNKSVD'
otherwise
error('unknown update method.')
end
sparseCoder = options.sparseCoder;
if isfield(options,'timeBudgetUpdate')
timeBudgetUpdate = options.timeBudgetUpdate;
options.numUpdateIter = [];
else
timeBudgetUpdate = [];
end
if isfield(options,'sparseCoderParams')
sparseCoderParams = options.sparseCoderParams;
else
sparseCoderParams = [];
end
if isfield(options,'verbosity')
VERBOSITY = options.verbosity;
else
VERBOSITY = 1;
end
[D,N] = size(X);
K = options.K;
L = options.L;
numIter = options.numIter;
updateType = options.updateType;
numUpdateIter = options.numUpdateIter;
if nargout > 2
E = zeros(numIter,1);
E_SC = zeros(numIter,1);
SCtime = zeros(numIter,1);
UDtime = zeros(numIter,1);
INFO = [];
end
if K >= N
H = diag(sqrt(sum(X.^2)));
H = [H; zeros(K-N,N)];
W = X;
W = W .* diag(1./sqrt(sum(W.^2)));
W = [W, ones(D,K-N) / D];
warning('too little data, trivial solution is returned.')
return
end
if isfield(options,'initType') && strcmp(options.initType, 'rand')
initType = 'rand';
else
initType = 'samples';
end
if isfield(options,'W')
if any(options.W(:)<0), error('options.W contains negative values.'); end
W = options.W;
else
if strcmp(initType, 'rand')
W = rand(D,K);
else
rp = randperm(N);
W = X(:,rp(1:K));
end
end
W = W * diag(1./sqrt(sum(W.^2)));
%%%-----------------------------------------------------------------%%%
for iter = 1:numIter
if VERBOSITY
fprintf('Iteration: %d ',iter);
end
%%% Sparse Coder Stage
tic
if isempty(sparseCoderParams)
H = sparseCoder(X,W,L);
else
H = sparseCoder(X,W,L,sparseCoderParams);
end
elapsedT = toc;
if nargout > 2
E_SC(iter) = norm(X-W*H,'fro');
SCtime(iter) = elapsedT;
end
if VERBOSITY
fprintf('SC-Error: %d ',norm(X-W*H,'fro')/norm(X,'fro'));
end
unusedIdx = all(H == 0,2);
numUnused = sum(unusedIdx);
W = W(:,~unusedIdx);
H = H(~unusedIdx,:);
%%% Update Stage
switch updateType
case 'MU'
if isempty(timeBudgetUpdate)
tic
for k = 1:numUpdateIter
W = W .* ((X*H') ./ (W*H*H' + 1e-12));
if k < numUpdateIter
H = H .* ((W'*X) ./ (W'*W*H + 1e-12));
end
end
elapsedT = toc;
else
tic
while toc < timeBudgetUpdate(iter)
W = W .* ((X*H') ./ (W*H*H' + 1e-12));
H = H .* ((W'*X) ./ (W'*W*H + 1e-12));
end
elapsedT = toc;
end
case 'ANLS_FC'
if isempty(timeBudgetUpdate)
tic
mask = H > 0;
for k = 1:numUpdateIter
%fprintf('update W\n')
Wt = sparseNNLS(X',H',[],[],K,K,W');
W = Wt';
if k < numUpdateIter
%fprintf('update H\n')
H = maskNNLS(X,W,[],[],mask,H);
end
end
elapsedT = toc;
else
tic
mask = H > 0;
while toc < timeBudgetUpdate(iter)
fprintf('.')
Wt = sparseNNLS(X',H',[],[],K,K,W');
W = Wt';
if toc >= timeBudgetUpdate(iter)
break
end
H = maskNNLS(X,W,[],[],mask,H);
end
fprintf('\n')
elapsedT = toc;
end
case 'ANLS_PG'
%%% no time budget variant here
if isempty(numUpdateIter)
error('time budget variant not implemented for ANLS via projected gradient.\n');
end
tic
mask = H > 0;
for k = 1:numUpdateIter
Wt = NNLS_PG(X',H',W',NNLS_PG_tolerance,NNLS_PG_maxIter);
W = Wt';
if k < numUpdateIter
H = NNLS_PG_mask(X,W,H,NNLS_PG_tolerance,NNLS_PG_maxIter,mask);
end
end
elapsedT = toc;
case 'NNKSVD'
K = size(W,2);
if isempty(timeBudgetUpdate)
tic
for k = 1:K
idx = H(k,:) > 0;
Etild = X(:,idx) - W(:,1:k-1) * H(1:k-1,idx) - W(:,k+1:end) * H(k+1:end,idx);
[U,S,V] = svds(Etild,1);
V = V'*S;
pu = max(U,0);
pv = max(V,0);
pu2 = max(-U,0);
pv2 = max(-V,0);
if norm(Etild - pu2 * pv2,'fro') < norm(Etild - pu * pv,'fro')
pu = pu2;
pv = pv2;
end
for l=1:numUpdateIter
pu = Etild * pv' / (pv*pv');
pv = pu' * Etild / (pu'*pu);
pu = max(pu,0);
pv = max(pv,0);
end
H(k,idx) = pv * norm(pu);
W(:,k) = pu / norm(pu);
end
elapsedT = toc;
else
elapsedT = 0;
rp = randperm(K);
for kcount = 1:K
tic
k = rp(kcount);
idx = H(k,:) > 0;
Etild = X(:,idx) - W(:,1:k-1) * H(1:k-1,idx) - W(:,k+1:end) * H(k+1:end,idx);
[U,S,V] = svds(Etild,1);
V = V'*S;
pu = max(U,0);
pv = max(V,0);
pu2 = max(-U,0);
pv2 = max(-V,0);
if norm(Etild - pu2 * pv2,'fro') < norm(Etild - pu * pv,'fro')
pu = pu2;
pv = pv2;
end
while toc < timeBudgetUpdate(iter) / K
pu = Etild * pv' / max((pv*pv'),1e-12);
pv = pu' * Etild / max((pu'*pu),1e-12);
pu = max(pu,0);
pv = max(pv,0);
end
H(k,idx) = pv * norm(pu);
W(:,k) = pu / norm(pu);
elapsedT = elapsedT + toc;
end
end
end
if nargout > 2
E(iter) = norm(X-W*H,'fro');
UDtime(iter) = elapsedT;
end
if VERBOSITY
fprintf('Update-Error: %d\n',norm(X-W*H,'fro')/norm(X,'fro'));
end
%%% Reinitialize
if strcmp(initType, 'rand')
W = [W,rand(D,numUnused)];
else
rp = randperm(N);
W = [W,X(:,rp(1:numUnused))];
end
W = W * diag(1./sqrt(sum(W.^2)));
end
if nargout > 2
INFO.E = E;
INFO.E_SC = E_SC;
INFO.SCtime = SCtime;
INFO.UDtime = UDtime;
end