Given partial observation of a low rank matrix, attempts to complete it. function low_rank_matrix_completion() This example demonstrates how to use the geometry factory for the embedded submanifold of fixed-rank matrices, fixedrankembeddedfactory. This geometry is described in the paper "Low-rank matrix completion by Riemannian optimization" Bart Vandereycken - SIAM Journal on Optimization, 2013. This can be a starting point for many optimization problems of the form: minimize f(X) such that rank(X) = k, size(X) = [m, n]. Note that the code is long because it showcases quite a few features of Manopt: most of the code is optional. Input: None. This example file generates random data. Output: None.

- fixedrankembeddedfactory Manifold struct to optimize fixed-rank matrices w/ an embedded geometry.
- conjugategradient Conjugate gradient minimization algorithm for Manopt.
- trustregions Riemannian trust-regions solver for optimization on manifolds.
- hessianspectrum Returns the eigenvalues of the (preconditioned) Hessian at x.

- function f = cost(X)
- function G = egrad(X)
- function ehess = euclidean_hessian(X, H)
- function t = linesearch_helper(X, H)

0001 function low_rank_matrix_completion() 0002 % Given partial observation of a low rank matrix, attempts to complete it. 0003 % 0004 % function low_rank_matrix_completion() 0005 % 0006 % This example demonstrates how to use the geometry factory for the 0007 % embedded submanifold of fixed-rank matrices, fixedrankembeddedfactory. 0008 % This geometry is described in the paper 0009 % "Low-rank matrix completion by Riemannian optimization" 0010 % Bart Vandereycken - SIAM Journal on Optimization, 2013. 0011 % 0012 % This can be a starting point for many optimization problems of the form: 0013 % 0014 % minimize f(X) such that rank(X) = k, size(X) = [m, n]. 0015 % 0016 % Note that the code is long because it showcases quite a few features of 0017 % Manopt: most of the code is optional. 0018 % 0019 % Input: None. This example file generates random data. 0020 % 0021 % Output: None. 0022 0023 % This file is part of Manopt and is copyrighted. See the license file. 0024 % 0025 % Main author: Nicolas Boumal, July 15, 2014 0026 % Contributors: Bart Vandereycken 0027 % 0028 % Change log: 0029 % 0030 0031 % Random data generation. First, choose the size of the problem. 0032 % We will complete a matrix of size mxn of rank k: 0033 m = 200; 0034 n = 500; 0035 k = 10; 0036 % Generate a random mxn matrix A of rank k 0037 L = randn(m, k); 0038 R = randn(n, k); 0039 A = L*R'; 0040 % Generate a random mask for observed entries: P(i, j) = 1 if the entry 0041 % (i, j) of A is observed, and 0 otherwise. 0042 fraction = 4 * k*(m+n-k)/(m*n); 0043 P = sparse(rand(m, n) <= fraction); 0044 % Hence, we know the nonzero entries in PA: 0045 PA = P.*A; 0046 0047 0048 0049 0050 0051 0052 0053 % Pick the manifold of matrices of size mxn of fixed rank k. 0054 problem.M = fixedrankembeddedfactory(m, n, k); 0055 0056 % Define the problem cost function. The input X is a structure with 0057 % fields U, S, V representing a rank k matrix as U*S*V'. 0058 % f(X) = 1/2 * || P.*(X-A) ||^2 0059 problem.cost = @cost; 0060 function f = cost(X) 0061 % Note that it is very much inefficient to explicitly construct the 0062 % matrix X in this way. Seen as we only need to know the entries 0063 % of Xmat corresponding to the mask P, it would be far more 0064 % efficient to compute those only. 0065 Xmat = X.U*X.S*X.V'; 0066 f = .5*norm( P.*Xmat - PA , 'fro')^2; 0067 end 0068 0069 % Define the Euclidean gradient of the cost function, that is, the 0070 % gradient of f(X) seen as a standard function of X. 0071 % nabla f(X) = P.*(X-A) 0072 problem.egrad = @egrad; 0073 function G = egrad(X) 0074 % Same comment here about Xmat. 0075 Xmat = X.U*X.S*X.V'; 0076 G = P.*Xmat - PA; 0077 end 0078 0079 % This is optional, but it's nice if you have it. 0080 % Define the Euclidean Hessian of the cost at X, along H, where H is 0081 % represented as a tangent vector: a structure with fields Up, Vp, M. 0082 % This is the directional derivative of nabla f(X) at X along Xdot: 0083 % nabla^2 f(X)[Xdot] = P.*Xdot 0084 problem.ehess = @euclidean_hessian; 0085 function ehess = euclidean_hessian(X, H) 0086 % The function tangent2ambient transforms H (a tangent vector) into 0087 % its equivalent ambient vector representation. The output is a 0088 % structure with fields U, S, V such that U*S*V' is an mxn matrix 0089 % corresponding to the tangent vector H. Note that there are no 0090 % additional guarantees about U, S and V. In particular, U and V 0091 % are not orthonormal. 0092 ambient_H = problem.M.tangent2ambient(X, H); 0093 Xdot = ambient_H.U*ambient_H.S*ambient_H.V'; 0094 % Same comment here about explicitly constructing the ambient 0095 % vector as an mxn matrix Xdot: we only need its entries 0096 % corresponding to the mask P, and this could be computed 0097 % efficiently. 0098 ehess = P.*Xdot; 0099 end 0100 0101 0102 0103 0104 % Check consistency of the gradient and the Hessian. Useful if you 0105 % adapt this example for a new cost function and you would like to make 0106 % sure there is no mistake. 0107 % warning('off', 'manopt:fixedrankembeddedfactory:exp'); 0108 % checkgradient(problem); pause; 0109 % checkhessian(problem); pause; 0110 0111 0112 0113 0114 0115 % Compute an initial guess. Points on the manifold are represented as 0116 % structures with three fields: U, S and V. U and V need to be 0117 % orthonormal, S needs to be diagonal. 0118 [U, S, V] = svds(PA, k); 0119 X0.U = U; 0120 X0.S = S; 0121 X0.V = V; 0122 0123 % Minimize the cost function using Riemannian trust-regions, starting 0124 % from the initial guess X0. 0125 X = trustregions(problem, X0); 0126 0127 % The reconstructed matrix is X, represented as a structure with fields 0128 % U, S and V. 0129 Xmat = X.U*X.S*X.V'; 0130 fprintf('||X-A||_F = %g\n', norm(Xmat - A, 'fro')); 0131 0132 0133 0134 0135 0136 0137 % Alternatively, we could decide to use a solver such as 0138 % steepestdescent or conjugategradient. These solvers need to solve a 0139 % line-search problem at each iteration. Standard line searches in 0140 % Manopt have generic purpose systems to do this. But for the problem 0141 % at hand, it so happens that we can rather accurately guess how far 0142 % the line-search should look, and it would be a waste to not use that. 0143 % Look up the paper referenced above for the mathematical explanation 0144 % of the code below. 0145 % 0146 % To tell Manopt about this special information, we specify the 0147 % linesearch hint function in the problem structure. Notice that this 0148 % is not the same thing as specifying a linesearch function in the 0149 % options structure. 0150 % 0151 % Both the SD and the CG solvers will detect that we 0152 % specify the hint function below, and they will use an appropriate 0153 % linesearch algorithm by default, as a result. Typically, they will 0154 % try the step t*H first, then if it does not satisfy an Armijo 0155 % criterion, they will decrease t geometrically until satisfaction or 0156 % failure. 0157 % 0158 % Just like the cost, egrad and ehess functions, the linesearch 0159 % function could use a store structure if you like. The present code 0160 % does not use the store structure, which means quite a bit of the 0161 % computations are made redundantly, and as a result a better method 0162 % could appear slower. See the Manopt tutorial about caching when you 0163 % are ready to switch from a proof-of-concept code to an efficient 0164 % code. 0165 % 0166 % The inputs are X (a point on the manifold) and H, a tangent vector at 0167 % X that is assumed to be a descent direction. That is, there exists a 0168 % positive t such that f(Retraction_X(tH)) < f(X). The function below 0169 % is supposed to output a "t" that it is a good "guess" at such a t. 0170 problem.linesearch = @linesearch_helper; 0171 function t = linesearch_helper(X, H) 0172 % Note that you would not usually need the Hessian for this. 0173 residual_omega = nonzeros(problem.egrad(X)); 0174 dir_omega = nonzeros(problem.ehess(X, H)); 0175 t = - dir_omega \ residual_omega ; 0176 end 0177 0178 % Notice that for this solver, the Hessian is not needed. 0179 [Xcg, xcost, info, options] = conjugategradient(problem, X0); %#ok<ASGLU> 0180 0181 fprintf('Take a look at the options that CG used:\n'); 0182 disp(options); 0183 fprintf('And see how many trials were made at each line search call:\n'); 0184 info_ls = [info.linesearch]; 0185 disp([info_ls.costevals]); 0186 0187 0188 0189 0190 0191 0192 0193 0194 fprintf('Try it again without the linesearch helper.\n'); 0195 0196 % Remove the linesearch helper from the problem structure. 0197 problem = rmfield(problem, 'linesearch'); 0198 0199 [Xcg, xcost, info, options] = conjugategradient(problem, X0); %#ok<ASGLU> 0200 0201 fprintf('Take a look at the options that CG used:\n'); 0202 disp(options); 0203 fprintf('And see how many trials were made at each line search call:\n'); 0204 info_ls = [info.linesearch]; 0205 disp([info_ls.costevals]); 0206 0207 0208 0209 0210 0211 0212 0213 % If the problem has a small enough dimension, we may (for analysis 0214 % purposes) compute the spectrum of the Hessian at a point X. This may 0215 % help in studying the conditioning of a problem. If you don't provide 0216 % the Hessian, Manopt will approximate the Hessian with finite 0217 % differences of the gradient and try to estimate its "spectrum" (it's 0218 % not a proper linear operator). This can give some intuition, but 0219 % should not be relied upon. 0220 if problem.M.dim() < 100 0221 fprintf('Computing the spectrum of the Hessian...'); 0222 s = hessianspectrum(problem, X); 0223 hist(s); 0224 end 0225 0226 0227 0228 0229 end

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