spm_svd.m

function [U,S,V] = spm_svd(X,U)
% Computationally efficient SVD (that can handle sparse arguments)
% FORMAT [U,S,V] = spm_svd(X,u)
% X    - (m x n) matrix
% u    - threshold (1 > u > 0) for normalized eigenvalues (default = 1e-6)
%      - a value of zero induces u = 64*eps
%
% U    - {m x p} singular vectors
% V    - {m x p} singular variates
% S    - {p x p} singular values
%__________________________________________________________________________
% Copyright (C) 1994-2011 Wellcome Trust Centre for Neuroimaging

% Karl Friston
% $Id: spm_svd.m 6110 2014-07-21 09:36:13Z karl $
# SPDX-License-Identifier: GPL-2.0

% default thresholds - preclude singular vectors with small singular values
%--------------------------------------------------------------------------
if nargin < 2, U = 1e-6; end
if U >= 1; U = U - 1e-6; end
if U <= 0; U = 64*eps;   end

% deal with sparse matrices
%--------------------------------------------------------------------------
[M,N] = size(X);
p     = find(any(X,2));
q     = find(any(X,1));
X     = X(p,q);

% SVD
%--------------------------------------------------------------------------
[i, j, s] = find(X);
[m, n]    = size(X);
if any(i - j)
    
    % off-leading diagonal elements - full SVD
    %----------------------------------------------------------------------
    X     = full(X);
    if m > n
        
        [v, S, v] = svd(X'*X,0);
        S         = sparse(S);
        s         = diag(S);
        j         = find(s*length(s)/sum(s) > U);
        v         = v(:,j);
        u         = spm_en(X*v);
        S         = sqrt(S(j,j));
        
    elseif m < n
        
        [u, S, u] = svd(X*X',0);
        S         = sparse(S);
        s         = diag(S);
        j         = find(s*length(s)/sum(s) > U);
        u         = u(:,j);
        v         = spm_en(X'*u);
        S         = sqrt(S(j,j));
        
    else
        
        [u, S, v] = svd(X,0);
        S         = sparse(S);
        s         = diag(S).^2;
        j         = find(s*length(s)/sum(s) > U);
        v         = v(:,j);
        u         = u(:,j);
        S         = S(j,j);
    end
    
else
    S             = sparse(1:n,1:n,s,m,n);
    u             = speye(m,n);
    v             = speye(m,n);
    [i, j]        = sort(-s);
    S             = S(j,j);
    v             = v(:,j);
    u             = u(:,j);
    s             = diag(S).^2;
    j             = find(s*length(s)/sum(s) > U);
    v             = v(:,j);
    u             = u(:,j);
    S             = S(j,j);
    
end

% replace in full matrices
%--------------------------------------------------------------------------
j      = length(j);
U      = sparse(M,j);
V      = sparse(N,j);
if j
    U(p,:) = u;
    V(q,:) = v;
end