Returns an orthonormal basis of tangent vectors in the Manopt framework. function orthobasis = tangentorthobasis(M, x) function orthobasis = tangentorthobasis(M, x, n) M is a Manopt manifold structure obtained from a factory. x is a point on the manifold M. n (optional) is the dimension of the random subspace to span; by default, n = M.dim() so that the returned basis spans the whole tangent space. orthobasis is a cell of n tangent vectors at x. With high probability, they form an orthonormal basis of the tangent space at x. If necessary, this can be checked by calling G = grammatrix(M, x, orthobasis) and verifying that norm(G - eye(size(G))) is close to zero. Note: if extra accuracy is required, it may help to re-orthogonalize the basis returned by this function once, as follows: B = tangentorthobasis(M, x, n); B = orthogonalize(M, x, B); See also: grammatrix orthogonalize lincomb plotprofile

- orthogonalize Orthonormalizes a basis of tangent vectors in the Manopt framework.

- getGradient Computes the gradient of the cost function at x.
- getGradientFD Computes an approx. of the gradient w/ finite differences of the cost.
- approxgradientFD Gradient approx. fnctn handle based on finite differences of the cost.
- hessianmatrix Computes a matrix which represents the Hessian in some tangent basis.

0001 function orthobasis = tangentorthobasis(M, x, n) 0002 % Returns an orthonormal basis of tangent vectors in the Manopt framework. 0003 % 0004 % function orthobasis = tangentorthobasis(M, x) 0005 % function orthobasis = tangentorthobasis(M, x, n) 0006 % 0007 % M is a Manopt manifold structure obtained from a factory. 0008 % x is a point on the manifold M. 0009 % n (optional) is the dimension of the random subspace to span; by default, 0010 % n = M.dim() so that the returned basis spans the whole tangent space. 0011 % 0012 % orthobasis is a cell of n tangent vectors at x. 0013 % With high probability, they form an orthonormal basis of the tangent 0014 % space at x. If necessary, this can be checked by calling 0015 % G = grammatrix(M, x, orthobasis) 0016 % and verifying that norm(G - eye(size(G))) is close to zero. 0017 % 0018 % Note: if extra accuracy is required, it may help to re-orthogonalize the 0019 % basis returned by this function once, as follows: 0020 % B = tangentorthobasis(M, x, n); 0021 % B = orthogonalize(M, x, B); 0022 % 0023 % See also: grammatrix orthogonalize lincomb plotprofile 0024 0025 % This file is part of Manopt: www.manopt.org. 0026 % Original author: Nicolas Boumal, April 28, 2016. 0027 % Contributors: 0028 % Change log: 0029 0030 dim = M.dim(); 0031 if ~exist('n', 'var') || isempty(n) 0032 n = dim; 0033 end 0034 assert(n >= 0 && n <= dim && n == round(n), ... 0035 'n must be an integer between 0 and M.dim().'); 0036 0037 basis = cell(n, 1); 0038 0039 % With high probability, n vectors taken at random in the tangent space 0040 % are linearly independent. 0041 for k = 1 : n 0042 basis{k} = M.randvec(x); 0043 end 0044 0045 % The Gram-Schmidt process transforms any n linearly independent 0046 % vectors into n orthonormal vectors spanning the same subspace. 0047 orthobasis = orthogonalize(M, x, basis); 0048 0049 end

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