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cp_als_trace.m
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cp_als_trace.m
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function [P,Uinit,output] = cp_als(X,R,varargin)
%CP_ALS Compute a CP decomposition of any type of tensor.
%
% M = CP_ALS(X,R) computes an estimate of the best rank-R
% CP model of a tensor X using an alternating least-squares
% algorithm. The input X can be a tensor, sptensor, ktensor, or
% ttensor. The result M is a ktensor.
%
% M = CP_ALS(X,R,'param',value,...) specifies optional parameters and
% values. Valid parameters and their default values are:
% 'tol' - Tolerance on difference in fit {1.0e-4}
% 'maxiters' - Maximum number of iterations {50}
% 'dimorder' - Order to loop through dimensions {1:ndims(A)}
% 'init' - Initial guess [{'random'}|'nvecs'|cell array]
% 'printitn' - Print fit every n iterations; 0 for no printing {1}
%
% [M,U0] = CP_ALS(...) also returns the initial guess.
%
% [M,U0,out] = CP_ALS(...) also returns additional output that contains
% the input parameters.
%
% Note: The "fit" is defined as 1 - norm(X-full(M))/norm(X) and is
% loosely the proportion of the data described by the CP model, i.e., a
% fit of 1 is perfect.
%
% NOTE: Updated in various minor ways per work of Phan Anh Huy. See Anh
% Huy Phan, Petr Tichavský, Andrzej Cichocki, On Fast Computation of
% Gradients for CANDECOMP/PARAFAC Algorithms, arXiv:1204.1586, 2012.
%
% Examples:
% X = sptenrand([5 4 3], 10);
% M = cp_als(X,2);
% M = cp_als(X,2,'dimorder',[3 2 1]);
% M = cp_als(X,2,'dimorder',[3 2 1],'init','nvecs');
% U0 = {rand(5,2),rand(4,2),[]}; %<-- Initial guess for factors of M
% [M,U0,out] = cp_als(X,2,'dimorder',[3 2 1],'init',U0);
% M = cp_als(X,2,out.params); %<-- Same params as previous run
%
% <a href="matlab:web(strcat('file://',...
% fullfile(getfield(what('tensor_toolbox'),'path'),'doc','html',...
% 'cp_als_doc.html')))">Documentation page for CP-ALS</a>
%
% See also KTENSOR, TENSOR, SPTENSOR, TTENSOR.
%
%MATLAB Tensor Toolbox. Copyright 2018, Sandia Corporation.
%% Extract number of dimensions and norm of X.
N = ndims(X);
normX = norm(X);
sz = size(X);
tsz = prod(sz);
if isa(X,'sptensor')
nnonzeros = nnz(X);
nzeros = tsz - nnonzeros;
info.nnonzeros = nnonzeros;
info.nzeros = nzeros;
end
% Save info
info.tsz = tsz;
%% Set algorithm parameters from input or by using defaults
params = inputParser;
params.addParameter('tol',1e-4,@isscalar);
params.addParameter('maxiters',50,@(x) isscalar(x) & x > 0);
params.addParameter('dimorder',1:N,@(x) isequal(sort(x),1:N));
params.addParameter('init', 'random', @(x) (iscell(x) || ismember(x,{'random','nvecs'})));
params.addParameter('printitn',1,@isscalar);
params.parse(varargin{:});
%% Copy from params object
fitchangetol = params.Results.tol;
maxiters = params.Results.maxiters;
dimorder = params.Results.dimorder;
init = params.Results.init;
printitn = params.Results.printitn;
%% Initialize trace and start timer
fit_trace = zeros(maxiters+1,1);
time_trace = zeros(maxiters+1,1);
main_time = tic;
%% Error checking
%% Set up and error checking on initial guess for U.
if iscell(init)
Uinit = init;
if numel(Uinit) ~= N
error('OPTS.init does not have %d cells',N);
end
for n = dimorder(2:end)
if ~isequal(size(Uinit{n}),[size(X,n) R])
error('OPTS.init{%d} is the wrong size',n);
end
end
else
% Observe that we don't need to calculate an initial guess for the
% first index in dimorder because that will be solved for in the first
% inner iteration.
if strcmp(init,'random')
Uinit = cell(N,1);
for n = dimorder(2:end)
Uinit{n} = rand(size(X,n),R);
end
elseif strcmp(init,'nvecs') || strcmp(init,'eigs')
Uinit = cell(N,1);
for n = dimorder(2:end)
Uinit{n} = nvecs(X,n,R);
end
else
error('The selected initialization method is not supported');
end
end
%% Set up for iterations - initializing U and the fit.
U = Uinit;
fit = 0;
fit_trace(1) = fit;
time_trace(1) = toc(main_time);
% Store the last MTTKRP result to accelerate fitness computation.
U_mttkrp = zeros(size(X, dimorder(end)), R);
%% Welcome message
if printitn > 0
fprintf('\n');
fprintf('CP_ALS (with trace):\n');
fprintf('\n');
fprintf('Tensor size: %s (%d total entries)\n', tt_size2str(size(X)), tsz);
if isa(X,'sptensor')
fprintf('Sparse tensor: %d (%.2g%%) Nonzeros and %d (%.2f%%) Zeros\n', nnonzeros, 100*nnonzeros/tsz, nzeros, 100*nzeros/tsz);
end
fprintf('Finding CP decomposition with R=%d\n', R);
fprintf('Fit change tolerance: %d\n', fitchangetol);
fprintf('Max iterations: %d\n', maxiters);
fprintf('\n');
end
%% Main Loop: Iterate until convergence
if (isa(X,'sptensor') || isa(X,'tensor')) && (exist('cpals_core','file') == 3)
%fprintf('Using C++ code\n');
[lambda,U] = cpals_core(X, Uinit, fitchangetol, maxiters, dimorder);
P = ktensor(lambda,U);
else
UtU = zeros(R,R,N);
for n = 1:N
if ~isempty(U{n})
UtU(:,:,n) = U{n}'*U{n};
end
end
for iter = 1:maxiters
fitold = fit;
% Iterate over all N modes of the tensor
for n = dimorder(1:end)
% Calculate Unew = X_(n) * khatrirao(all U except n, 'r').
Unew = mttkrp(X,U,n);
% Save the last MTTKRP result for fitness check.
if n == dimorder(end)
U_mttkrp = Unew;
end
% Compute the matrix of coefficients for linear system
Y = prod(UtU(:,:,[1:n-1 n+1:N]),3);
Unew = Unew / Y;
if issparse(Unew)
Unew = full(Unew); % for the case R=1
end
% Normalize each vector to prevent singularities in coefmatrix
if iter == 1
lambda = sqrt(sum(Unew.^2,1))'; %2-norm
else
lambda = max( max(abs(Unew),[],1), 1 )'; %max-norm
end
Unew = bsxfun(@rdivide, Unew, lambda');
U{n} = Unew;
UtU(:,:,n) = U{n}'*U{n};
end
P = ktensor(lambda,U);
% This is equivalent to innerprod(X,P).
iprod = sum(sum(P.U{dimorder(end)} .* U_mttkrp) .* lambda');
if normX == 0
fit = norm(P)^2 - 2 * iprod;
else
normresidual = sqrt( normX^2 + norm(P)^2 - 2 * iprod );
fit = 1 - (normresidual / normX); %fraction explained by model
end
fitchange = abs(fitold - fit);
% Check for convergence
if (iter > 1) && (fitchange < fitchangetol)
flag = 0;
else
flag = 1;
end
fit_trace(iter+1) = fit;
time_trace(iter+1) = toc(main_time);
if (mod(iter,printitn)==0) || ((printitn>0) && (flag==0))
fprintf(' Iter %2d: f = %e f-delta = %7.1e time = %.1f seconds \n', iter, fit, fitchange, time_trace(iter+1) - time_trace(iter));
end
% Check for convergence
if (flag == 0)
break;
end
end
end
%% Clean up final result
% Arrange the final tensor so that the columns are normalized.
P = arrange(P);
% Fix the signs
P = fixsigns(P);
if printitn>0
if normX == 0
fit = norm(P)^2 - 2 * innerprod(X,P);
else
normresidual = sqrt( normX^2 + norm(P)^2 - 2 * innerprod(X,P) );
fit = 1 - (normresidual / normX); %fraction explained by model
end
fprintf(' Final f = %e \n', fit);
end
output = struct;
output.params = params.Results;
output.iters = iter;
output.fit = fit;
output.time_trace = time_trace;
output.fit_trace = fit_trace;
output.normX = normX;