centeredmatrixfactory

 Linear manifold struct. for optimization over matrices with centered cols

 function M = centeredmatrixfactory(m, n)
 function M = centeredmatrixfactory(m, n, 'cols')
 function M = centeredmatrixfactory(m, n, 'rows')

 Returns M, a structure for Manopt describing the Euclidean space of
 m-by-n matrices whose columns sum to zero (or whose rows sum to zero,
 if 'rows' is passed as last input).

 The metric is the standard Frobenius distance and associated trace inner
 product. Matrices on M, denoted by X, have size mxn and obey
 X*ones(n, 1) = 0 (centered columns) or ones(1, m)*X = 0 (centered rows).

 See also: euclideanfactory

0001 function M = centeredmatrixfactory(m, n, rows_or_cols)
0002 % Linear manifold struct. for optimization over matrices with centered cols
0003 %
0004 % function M = centeredmatrixfactory(m, n)
0005 % function M = centeredmatrixfactory(m, n, 'cols')
0006 % function M = centeredmatrixfactory(m, n, 'rows')
0007 %
0008 % Returns M, a structure for Manopt describing the Euclidean space of
0009 % m-by-n matrices whose columns sum to zero (or whose rows sum to zero,
0010 % if 'rows' is passed as last input).
0011 %
0012 % The metric is the standard Frobenius distance and associated trace inner
0013 % product. Matrices on M, denoted by X, have size mxn and obey
0014 % X*ones(n, 1) = 0 (centered columns) or ones(1, m)*X = 0 (centered rows).
0015 %
0016 % See also: euclideanfactory
0017 
0018 % This file is part of Manopt: www.manopt.org.
0019 % Original author: Nicolas Boumal, July 3, 2015.
0020 % Contributors:
0021 % Change log:
0022 %
0023 %   Jan. 6, 2017 (NB):
0024 %       M.tangent = M.proj now, instead of being identity. This is notably
0025 %       necessary so that checkgradient will pick up on gradients that do
0026 %       not lie in the appropriate tangent space.
0027 
0028     if ~exist('rows_or_cols', 'var') || isempty(rows_or_cols)
0029         rows_or_cols = 'cols';
0030     end
0031     
0032     % Define a centering operator: it subtracts the mean column or row.
0033     switch lower(rows_or_cols)
0034         case 'cols'
0035             center = @(X) bsxfun(@minus, X, mean(X, 2));
0036             M.dim = @() m*n - m;
0037         case 'rows'
0038             center = @(X) bsxfun(@minus, X, mean(X, 1));
0039             M.dim = @() m*n - n;
0040         otherwise
0041             error('The third input must be either ''rows'' or ''cols''.');
0042     end
0043     
0044     % This is a non-standard function to have in a Manopt manifold.
0045     % It is included because it might be helpful in some situations.
0046     M.center = center;
0047 
0048     M.name = @() sprintf('Space of size %d x %d matrices with centered %s', ...
0049                          m, n, lower(rows_or_cols));
0050     
0051     M.inner = @(x, d1, d2) d1(:).'*d2(:);
0052     
0053     M.norm = @(x, d) norm(d, 'fro');
0054     
0055     M.dist = @(x, y) norm(x-y, 'fro');
0056     
0057     M.typicaldist = @() sqrt(M.dim());
0058     
0059     M.proj = @(x, d) center(d);
0060     
0061     M.egrad2rgrad = M.proj;
0062     
0063     M.ehess2rhess = @(x, eg, eh, d) center(eh);
0064     
0065     M.tangent = M.proj;
0066     
0067     M.exp = @exp;
0068     function y = exp(x, d, t)
0069         if nargin == 3
0070             y = x + t*d;
0071         else
0072             y = x + d;
0073         end
0074     end
0075     
0076     M.retr = M.exp;
0077     
0078     M.log = @(x, y) y-x;
0079 
0080     M.hash = @(x) ['z' hashmd5(x(:))];
0081     
0082     M.randvec = @(X) randvec();
0083     function U = randvec()
0084         U = center(randn(m, n));
0085         U = U / norm(U, 'fro');
0086     end
0087     
0088     M.rand = @() center(randn(m, n));
0089     
0090     M.lincomb = @matrixlincomb;
0091     
0092     M.zerovec = @(x) zeros(m, n);
0093     
0094     M.transp = @(x1, x2, d) d;
0095     
0096     M.pairmean = @(x1, x2) .5*(x1+x2);
0097     
0098     M.vec = @(x, u_mat) u_mat(:);
0099     M.mat = @(x, u_vec) reshape(u_vec, [m, n]);
0100     M.vecmatareisometries = @() true;
0101 
0102 end