spherecomplexfactory

 Returns a manifold struct to optimize over unit-norm complex matrices.

 function M = spherecomplexfactory(n)
 function M = spherecomplexfactory(n, m)

 Manifold of n-by-m complex matrices of unit Frobenius norm.
 By default, m = 1, which corresponds to the unit sphere in C^n. The
 metric is such that the sphere is a Riemannian submanifold of the space
 of 2nx2m real matrices with the usual trace inner product, i.e., the
 usual metric.
 
 See also: spherefactory

0001 function M = spherecomplexfactory(n, m)
0002 % Returns a manifold struct to optimize over unit-norm complex matrices.
0003 %
0004 % function M = spherecomplexfactory(n)
0005 % function M = spherecomplexfactory(n, m)
0006 %
0007 % Manifold of n-by-m complex matrices of unit Frobenius norm.
0008 % By default, m = 1, which corresponds to the unit sphere in C^n. The
0009 % metric is such that the sphere is a Riemannian submanifold of the space
0010 % of 2nx2m real matrices with the usual trace inner product, i.e., the
0011 % usual metric.
0012 %
0013 % See also: spherefactory
0014 
0015 % This file is part of Manopt: www.manopt.org.
0016 % Original author: Nicolas Boumal, Dec. 30, 2012.
0017 % Contributors:
0018 % Change log:
0019 %
0020 %   Sep. 4, 2014 (NB):
0021 %       Added ehess2rhess.
0022 %
0023 %   April 7, 2015 (NB):
0024 %       Added vec/mat pair (for use with hessianspectrum, for example).
0025 %
0026 %   April 13, 2015 (NB):
0027 %       Added logarithm
0028 %
0029 %   Oct. 8, 2016 (NB)
0030 %       Code for exponential was simplified to only treat the zero vector
0031 %       as a particular case.
0032 %
0033 %   Oct. 22, 2016 (NB)
0034 %       Distance function dist now significantly more accurate for points
0035 %       within 1e-7 and less from each other.
0036 
0037     
0038     if ~exist('m', 'var')
0039         m = 1;
0040     end
0041 
0042     if m == 1
0043         M.name = @() sprintf('Complex sphere S^%d', n-1);
0044     else
0045         M.name = @() sprintf('Unit F-norm %dx%d complex matrices', n, m);
0046     end
0047     
0048     M.dim = @() 2*(n*m)-1;
0049     
0050     M.inner = @(x, d1, d2) real(d1(:)'*d2(:));
0051     
0052     M.norm = @(x, d) norm(d, 'fro');
0053     
0054     M.dist = @(x, y) real(2*asin(.5*norm(x - y, 'fro')));
0055     
0056     M.typicaldist = @() pi;
0057     
0058     M.proj = @(x, d) reshape(d(:) - x(:)*(real(x(:)'*d(:))), n, m);
0059     
0060     % For Riemannian submanifolds, converting a Euclidean gradient into a
0061     % Riemannian gradient amounts to an orthogonal projection.
0062     M.egrad2rgrad = M.proj;
0063     
0064     M.ehess2rhess = @ehess2rhess;
0065     function rhess = ehess2rhess(x, egrad, ehess, u)
0066         rhess = M.proj(x, ehess) - real((x(:)'*egrad(:)))*u;
0067     end
0068     
0069     M.tangent = M.proj;
0070     
0071     M.exp = @exponential;
0072     
0073     M.retr = @retraction;
0074 
0075     M.log = @logarithm;
0076     function v = logarithm(x1, x2)
0077         v = M.proj(x1, x2 - x1);
0078         di = M.dist(x1, x2);
0079         % If the two points are "far apart", correct the norm.
0080         if di > 1e-6
0081             nv = norm(v, 'fro');
0082             v = v * (di / nv);
0083         end
0084     end
0085     
0086     M.hash = @(x) ['z' hashmd5([real(x(:)) ; imag(x(:))])];
0087     
0088     M.rand = @() random(n, m);
0089     
0090     M.randvec = @(x) randomvec(n, m, x);
0091     
0092     M.lincomb = @matrixlincomb;
0093     
0094     M.zerovec = @(x) zeros(n, m);
0095     
0096     M.transp = @(x1, x2, d) M.proj(x2, d);
0097     
0098     M.pairmean = @pairmean;
0099     function y = pairmean(x1, x2)
0100         y = x1+x2;
0101         y = y / norm(y, 'fro');
0102     end
0103 
0104     mn = m*n;
0105     M.vec = @(x, u_mat) [real(u_mat(:)) ; imag(u_mat(:))];
0106     M.mat = @(x, u_vec) reshape(u_vec(1:mn), m, n) + 1i*reshape(u_vec((mn+1):end), m, n);
0107     M.vecmatareisometries = @() true;
0108 
0109 end
0110 
0111 % Exponential on the sphere
0112 function y = exponential(x, d, t)
0113 
0114     if nargin == 2
0115         % t = 1;
0116         td = d;
0117     else
0118         td = t*d;
0119     end
0120     
0121     nrm_td = norm(td, 'fro');
0122     
0123     if nrm_td > 0
0124         y = x*cos(nrm_td) + td*(sin(nrm_td)/nrm_td);
0125     else
0126         y = x;
0127     end
0128 
0129 end
0130 
0131 % Retraction on the sphere
0132 function y = retraction(x, d, t)
0133 
0134     if nargin == 2
0135         t = 1;
0136     end
0137     
0138     y = x+t*d;
0139     y = y/norm(y, 'fro');
0140 
0141 end
0142 
0143 % Uniform random sampling on the sphere.
0144 function x = random(n, m)
0145 
0146     x = randn(n, m) + 1i*randn(n, m);
0147     x = x/norm(x, 'fro');
0148 
0149 end
0150 
0151 % Random normalized tangent vector at x.
0152 function d = randomvec(n, m, x)
0153 
0154     d = randn(n, m) + 1i*randn(n, m);
0155     d = reshape(d(:) - x(:)*(real(x(:)'*d(:))), n, m);
0156     d = d / norm(d, 'fro');
0157 
0158 end