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 function M = positivefactory(m, n) % Manifold of m-by-n matrices with positive entries; scale invariant metric % % function M = positivefactory(m) % function M = positivefactory(m, n) % % A point X on the manifold M is represented as a matrix X of size mxn with % all individual entries real, strictly positive. By default, n = 1. % % A tangent vector at X is represented as a matrix of the same size as X. % Entries of tangent vectors are free (in particular, not necessarily % positive.) % % The Riemannian metric for each individual entry is the bi-invariant % metric for positive scalars, as a particular case of the bi-invariant % metric for positive definite matrices studied in Chapter 6 of the book % % "Positive definite matrices" by Rajendra Bhatia, % Princeton University Press, 2007. % % The Riemannian structure of M is obtained as the Cartesian product of the % geometry for mxn positive real numbers. % % It should be stressed that matrices with one or more zero entries do not % belong to this manifold: they appear to be infinitely far away as a % result of the metric scaling like X.^(-1). Thus, if the solutions of an % optimization problem have entries equal to zero, these solutions are not % attainable on the manifold, which is likely to create serious numerical % issues. This geometry is best used when the solutions of the optimization % problem are indeed entry-wise positive, yet may have very different % scales (with some entries being very small, and some entries being very % large, relatively.) % % See also: sympositivedefinitefactory % This file is part of Manopt: www.manopt.org. % Original author: Bamdev Mishra, Dec 03, 2017. if ~exist('n', 'var') || isempty(n) n = 1; end M.name = @() sprintf('Element-wise positive %dx%d matrices', m, n); M.dim = @() m*n; % The metric is the scale invariant metric for scalars. M.inner = @myinner; function innerproduct = myinner(X, eta, zeta) innerproduct = (eta(:)./X(:))'*(zeta(:)./X(:)); end M.norm = @(X, eta) sqrt(myinner(X, eta, eta)); M.dist = @(X, Y) norm(log(Y./X), 'fro'); M.typicaldist = @() sqrt(m*n); M.egrad2rgrad = @egrad2rgrad; function eta = egrad2rgrad(X, eta) eta = X.*(eta).*X; end M.ehess2rhess = @ehess2rhess; function Hess = ehess2rhess(X, egrad, ehess, eta) % Directional derivatives of the Riemannian gradient Hess = X.*(ehess).*X + 2*(eta.*(egrad).*X); % Correction factor for the non-constant metric Hess = Hess - (eta.*(egrad).*X); end % Since this manifold is an open subset of R^(nxm), the tangent space % at any X on M is all of R^(nxm). M.proj = @(X, eta) eta; M.tangent = M.proj; M.tangent2ambient = @(X, eta) eta; M.retr = @exponential; M.exp = @exponential; function Y = exponential(X, eta, t) if nargin < 3 t = 1.0; end % It is unclear whether this is the numerically most stable way to % implement this operation. If you run into trouble with this % factory, please get in touch on the forum. Y = (X.*(exp((t*eta)./X))); end M.log = @logarithm; function H = logarithm(X, Y) % Same comment about numerical stability as for exp. H = (X.*(log(Y./X))); end M.hash = @(X) ['z' hashmd5(X(:))]; % Generate a random element-wise positive matrix following a % certain distribution. The particular choice of a distribution is of % course arbitrary, and specific applications might require different % ones. M.rand = @random; function X = random() X = exp(randn(m, n)); end % Generate a uniformly random unit-norm tangent vector at X. M.randvec = @randomvec; function eta = randomvec(X) eta = randn(m, n).*X; nrm = M.norm(X, eta); eta = eta / nrm; end M.lincomb = @matrixlincomb; M.zerovec = @(X) zeros(m, n); M.transp = @(X1, X2, eta) eta; % For reference, a proper vector transport is given here, following % work by Sra and Hosseini: "Conic geometric optimisation on the % manifold of positive definite matrices". % This is not used by default. To force the use of this transport, % execute "M.transp = M.paralleltransp;" on your M returned by the % present factory. M.paralleltransp = @parallel_transport; function zeta = parallel_transport(X, Y, eta) zeta = eta.*Y./X; end % vec and mat are not isometries, because of the unusual inner metric. M.vec = @(X, U) U(:); M.mat = @(X, u) reshape(u, m, n); M.vecmatareisometries = @() true; end