fixedrankMNquotientfactory

 Manifold of m-by-n matrices of rank k with two factor quotient geometry.

 function M = fixedrankMNquotientfactory(m, n, k)

 This follows the quotient geometry described in the following paper:
 P.-A. Absil, L. Amodei and G. Meyer,
 "Two Newton methods on the manifold of fixed-rank matrices endowed
  with Riemannian quotient geometries", arXiv, 2012.

 Paper link: http://arxiv.org/abs/1209.0068

 A point X on the manifold is represented as a structure with two
 fields: M and N. The matrix M (mxk) is orthonormal, while the matrix N
 (nxk) is full-rank such that X = M*N';

 Tangent vectors are represented as a structure with two fields (M, N).

 Please cite the Manopt paper as well as the research paper:
     @Article{absil2014fixedrank,
       Title   = {Two Newton methods on the manifold of fixed-rank matrices endowed with Riemannian quotient geometries},
       Author  = {Absil, P.-A. and Amodei, L. and Meyer, G.},
       Journal = {Computational Statistics},
       Year    = {2014},
       Number  = {3-4},
       Pages   = {569--590},
       Volume  = {29},
       Doi     = {10.1007/s00180-013-0441-6}
     }

0001 function M = fixedrankMNquotientfactory(m, n, k)
0002 % Manifold of m-by-n matrices of rank k with two factor quotient geometry.
0003 %
0004 % function M = fixedrankMNquotientfactory(m, n, k)
0005 %
0006 % This follows the quotient geometry described in the following paper:
0007 % P.-A. Absil, L. Amodei and G. Meyer,
0008 % "Two Newton methods on the manifold of fixed-rank matrices endowed
0009 %  with Riemannian quotient geometries", arXiv, 2012.
0010 %
0011 % Paper link: http://arxiv.org/abs/1209.0068
0012 %
0013 % A point X on the manifold is represented as a structure with two
0014 % fields: M and N. The matrix M (mxk) is orthonormal, while the matrix N
0015 % (nxk) is full-rank such that X = M*N';
0016 %
0017 % Tangent vectors are represented as a structure with two fields (M, N).
0018 %
0019 % Please cite the Manopt paper as well as the research paper:
0020 %     @Article{absil2014fixedrank,
0021 %       Title   = {Two Newton methods on the manifold of fixed-rank matrices endowed with Riemannian quotient geometries},
0022 %       Author  = {Absil, P.-A. and Amodei, L. and Meyer, G.},
0023 %       Journal = {Computational Statistics},
0024 %       Year    = {2014},
0025 %       Number  = {3-4},
0026 %       Pages   = {569--590},
0027 %       Volume  = {29},
0028 %       Doi     = {10.1007/s00180-013-0441-6}
0029 %     }
0030 
0031 % This file is part of Manopt: www.manopt.org.
0032 % Original author: Nicolas Boumal, Dec. 30, 2012.
0033 % Contributors:
0034 % Change log:
0035 %     NB, April 17, 2018: added M.tangent
0036 %     NB, April 18, 2018: removed lyap dependency
0037     
0038     
0039     M.name = @() sprintf('MN'' quotient manifold of %dx%d matrices of rank %d', m, n, k);
0040     
0041     M.dim = @() (m+n-k)*k;
0042     
0043     % Choice of the metric is motivated by the symmetry present in the
0044     % space.
0045     M.inner = @(X, eta, zeta) eta.M(:).'*zeta.M(:) + eta.N(:).'*zeta.N(:);
0046     
0047     M.norm = @(X, eta) sqrt(M.inner(X, eta, eta));
0048     
0049     M.dist = @(x, y) error('fixedrankMNquotientfactory.dist not implemented yet.');
0050     
0051     M.typicaldist = @() 10*k;
0052     
0053     symm = @(X) .5*(X+X');
0054     stiefel_proj = @(M, H) H - M*symm(M'*H);
0055     
0056     M.egrad2rgrad = @egrad2rgrad;
0057     function eta = egrad2rgrad(X, eta)
0058         eta.M = stiefel_proj(X.M, eta.M);
0059     end
0060     
0061     M.ehess2rhess = @ehess2rhess;
0062     function Hess = ehess2rhess(X, egrad, ehess, eta)
0063         
0064         % Directional derivative of the Riemannian gradient.
0065         Hess.M = ehess.M - eta.M*symm(X.M'*egrad.M);
0066         Hess.M = stiefel_proj(X.M, Hess.M);
0067         
0068         Hess.N = ehess.N;
0069         
0070         % Projection onto the horizontal space.
0071         Hess = M.proj(X, Hess);
0072     end
0073     
0074     
0075     M.proj = @projection;
0076     function etaproj = projection(X, eta)
0077         
0078         % Start by projecting the vector from Rmp x Rnp to the tangent
0079         % space to the total space, that is, eta.M should be in the
0080         % tangent space to Stiefel at X.M and eta.N is arbitrary.
0081         eta.M = stiefel_proj(X.M, eta.M);
0082         
0083         % Now project from the tangent space to the horizontal space, that
0084         % is, take care of the quotient.
0085         
0086         % First solve a Sylvester equation (A symm., B skew-symm.)
0087         A = X.N'*X.N + eye(k);
0088         B = eta.M'*X.M + eta.N'*X.N;
0089         B = B-B';
0090         omega = lyapunov_symmetric(A, B);
0091         
0092         % And project along the vertical space to the horizontal space.
0093         etaproj.M = eta.M + X.M*omega;
0094         etaproj.N = eta.N + X.N*omega;
0095         
0096     end
0097     
0098     M.tangent = M.proj;
0099     
0100     M.exp = @exponential;
0101     function Y = exponential(X, eta, t)
0102         if nargin < 3
0103             t = 1.0;
0104         end
0105         
0106         A = t*X.M'*eta.M;
0107         S = t^2*eta.M'*eta.M;
0108         Y.M = [X.M t*eta.M]*expm([A -S ; eye(k) A])*eye(2*k, k)*expm(-A);
0109         
0110         % re-orthonormalize (seems necessary from time to time).
0111         [Q R] = qr(Y.M, 0);
0112         Y.M = Q * diag(sign(diag(R)));
0113         
0114         Y.N = X.N + t*eta.N;
0115         
0116     end
0117     
0118     % Factor M lives on the Stiefel manifold, hence we will reuse its
0119     % random generator.
0120     stiefelm = stiefelfactory(m, k);
0121     
0122     M.retr = @retraction;
0123     function Y = retraction(X, eta, t)
0124         if nargin < 3
0125             t = 1.0;
0126         end
0127         
0128         Y.M = uf(X.M + t*eta.M); % This is a valid retraction
0129         Y.N = X.N + t*eta.N;
0130     end
0131     
0132     M.hash = @(X) ['z' hashmd5([X.M(:) ; X.N(:)])];
0133     
0134     M.rand = @random;
0135     function X = random()
0136         X.M = stiefelm.rand();
0137         X.N = randn(n, k);
0138     end
0139     
0140     M.randvec = @randomvec;
0141     function eta = randomvec(X)
0142         eta.M = randn(m, k);
0143         eta.N = randn(n, k);
0144         eta = projection(X, eta);
0145         nrm = M.norm(X, eta);
0146         eta.M = eta.M / nrm;
0147         eta.N = eta.N / nrm;
0148     end
0149     
0150     M.lincomb = @lincomb;
0151     
0152     M.zerovec = @(X) struct('M', zeros(m, k), 'N', zeros(n, k));
0153     
0154     M.transp = @(x1, x2, d) projection(x2, d);
0155     
0156 end
0157 
0158 
0159 % Linear combination of tangent vectors
0160 function d = lincomb(x, a1, d1, a2, d2) %#ok<INMSL>
0161     
0162     if nargin == 3
0163         d.M = a1*d1.M;
0164         d.N = a1*d1.N;
0165     elseif nargin == 5
0166         d.M = a1*d1.M + a2*d2.M;
0167         d.N = a1*d1.N + a2*d2.N;
0168     else
0169         error('Bad use of fixedrankMNquotientfactory.lincomb.');
0170     end
0171     
0172 end
0173 
0174 
0175 function A = uf(A)
0176     [L, unused, R] = svd(A, 0);
0177     A = L*R';
0178 end