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.
- disp DISP Display TT/MPS tensor.
- norm NORM Norm of a TT/MPS tensor.
- disp DISP Display TT/MPS block-mu tensor.
- norm NORM Norm of a TT/MPS block-mu tensor.
- disp DISP Display TT/MPS operator.
- disp DISP Display TT/MPS operator.
- 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 % Xiaowen Jiang Aug. 20, 2021 0031 % Added AD to compute the egrad and the ehess 0032 0033 % Random data generation. First, choose the size of the problem. 0034 % We will complete a matrix of size mxn of rank k. 0035 m = 200; 0036 n = 200; 0037 k = 5; 0038 % Generate a random mxn matrix A of rank k 0039 L = randn(m, k); 0040 R = randn(n, k); 0041 A = L*R'; 0042 % Generate a random mask for observed entries: P(i, j) = 1 if the entry 0043 % (i, j) of A is observed, and 0 otherwise. 0044 fraction = 4 * k*(m+n-k)/(m*n); 0045 P = sparse(rand(m, n) <= fraction); 0046 % Hence, we know the nonzero entries in PA: 0047 PA = P.*A; 0048 0049 0050 0051 0052 0053 0054 0055 % Pick the manifold of matrices of size mxn of fixed rank k. 0056 problem.M = fixedrankembeddedfactory(m, n, k); 0057 0058 % Define the problem cost function. The input X is a structure with 0059 % fields U, S, V representing a rank k matrix as U*S*V'. 0060 % f(X) = 1/2 * || P.*(X-A) ||^2 0061 problem.cost = @cost; 0062 function f = cost(X) 0063 % Note that it is very much inefficient to explicitly construct the 0064 % matrix X in this way. Seen as we only need to know the entries 0065 % of Xmat corresponding to the mask P, it would be far more 0066 % efficient to compute those only. 0067 Xmat = X.U*X.S*X.V'; 0068 f = .5*norm( P.*Xmat - PA , 'fro')^2; 0069 end 0070 0071 % Define the Euclidean gradient of the cost function, that is, the 0072 % gradient of f(X) seen as a standard function of X. 0073 % nabla f(X) = P.*(X-A) 0074 problem.egrad = @egrad; 0075 function G = egrad(X) 0076 % Same comment here about Xmat. 0077 Xmat = X.U*X.S*X.V'; 0078 G = P.*Xmat - PA; 0079 end 0080 0081 % This is optional, but it's nice if you have it. 0082 % Define the Euclidean Hessian of the cost at X, along H, where H is 0083 % represented as a tangent vector: a structure with fields Up, Vp, M. 0084 % This is the directional derivative of nabla f(X) at X along Xdot: 0085 % nabla^2 f(X)[Xdot] = P.*Xdot 0086 problem.ehess = @euclidean_hessian; 0087 function ehess = euclidean_hessian(X, H) 0088 % The function tangent2ambient transforms H (a tangent vector) into 0089 % its equivalent ambient vector representation. The output is a 0090 % structure with fields U, S, V such that U*S*V' is an mxn matrix 0091 % corresponding to the tangent vector H. Note that there are no 0092 % additional guarantees about U, S and V. In particular, U and V 0093 % are not orthonormal. 0094 ambient_H = problem.M.tangent2ambient(X, H); 0095 Xdot = ambient_H.U*ambient_H.S*ambient_H.V'; 0096 % Same comment here about explicitly constructing the ambient 0097 % vector as an mxn matrix Xdot: we only need its entries 0098 % corresponding to the mask P, and this could be computed 0099 % efficiently. 0100 ehess = P.*Xdot; 0101 end 0102 0103 % An alternative way to compute the egrad and the ehess is to use 0104 % automatic differentiation provided in the deep learning toolbox (slower) 0105 % Notice that the function norm is not supported for AD so far. 0106 % Replace norm(...,'fro')^2 with cnormsqfro described in the file 0107 % manoptADhelp.m. Also, operations between sparse matrices and dlarrays 0108 % are not supported for now. convert PA and P into full matrices. 0109 % PA_full = full(PA); 0110 % P_full = full(P); 0111 % problem.cost = @cost_AD; 0112 % function f = cost_AD(X) 0113 % Xmat = X.U*X.S*X.V'; 0114 % f = .5*cnormsqfro(P_full.*Xmat - PA_full); 0115 % end 0116 0117 % Computating the ehess for the set of fixed-rank matrices with 0118 % an embedded geometry is currently not supported. 0119 % Call manoptAD to automatically obtain the grad 0120 % problem = manoptAD(problem); 0121 0122 0123 % Check consistency of the gradient and the Hessian. Useful if you 0124 % adapt this example for a new cost function and you would like to make 0125 % sure there is no mistake. 0126 % warning('off', 'manopt:fixedrankembeddedfactory:exp'); 0127 % checkgradient(problem); pause; 0128 % checkhessian(problem); pause; 0129 0130 0131 0132 0133 0134 % Compute an initial guess. Points on the manifold are represented as 0135 % structures with three fields: U, S and V. U and V need to be 0136 % orthonormal, S needs to be diagonal. 0137 [U, S, V] = svds(PA, k); 0138 X0.U = U; 0139 X0.S = S; 0140 X0.V = V; 0141 0142 % Minimize the cost function using Riemannian trust-regions, starting 0143 % from the initial guess X0. 0144 X = trustregions(problem, X0); 0145 0146 % The reconstructed matrix is X, represented as a structure with fields 0147 % U, S and V. 0148 Xmat = X.U*X.S*X.V'; 0149 fprintf('||X-A||_F = %g\n', norm(Xmat - A, 'fro')); 0150 0151 0152 0153 0154 0155 0156 % Alternatively, we could decide to use a solver such as 0157 % steepestdescent or conjugategradient. These solvers need to solve a 0158 % line-search problem at each iteration. Standard line searches in 0159 % Manopt have generic purpose systems to do this. But for the problem 0160 % at hand, it so happens that we can rather accurately guess how far 0161 % the line-search should look, and it would be a waste to not use that. 0162 % Look up the paper referenced above for the mathematical explanation 0163 % of the code below. 0164 % 0165 % To tell Manopt about this special information, we specify the 0166 % linesearch hint function in the problem structure. Notice that this 0167 % is not the same thing as specifying a linesearch function in the 0168 % options structure. 0169 % 0170 % Both the SD and the CG solvers will detect that we 0171 % specify the hint function below, and they will use an appropriate 0172 % linesearch algorithm by default, as a result. Typically, they will 0173 % try the step t*H first, then if it does not satisfy an Armijo 0174 % criterion, they will decrease t geometrically until satisfaction or 0175 % failure. 0176 % 0177 % Just like the cost, egrad and ehess functions, the linesearch 0178 % function could use a store structure if you like. The present code 0179 % does not use the store structure, which means quite a bit of the 0180 % computations are made redundantly, and as a result a better method 0181 % could appear slower. See the Manopt tutorial about caching when you 0182 % are ready to switch from a proof-of-concept code to an efficient 0183 % code. 0184 % 0185 % The inputs are X (a point on the manifold) and H, a tangent vector at 0186 % X that is assumed to be a descent direction. That is, there exists a 0187 % positive t such that f(Retraction_X(tH)) < f(X). The function below 0188 % is supposed to output a "t" that it is a good "guess" at such a t. 0189 problem.linesearch = @linesearch_helper; 0190 function t = linesearch_helper(X, H) 0191 % Note that you would not usually need the Hessian for this. 0192 residual_omega = nonzeros(problem.egrad(X)); 0193 dir_omega = nonzeros(problem.ehess(X, H)); 0194 t = - dir_omega \ residual_omega ; 0195 end 0196 0197 % Notice that for this solver, the Hessian is not needed. 0198 [Xcg, xcost, info, options] = conjugategradient(problem, X0); %#ok<ASGLU> 0199 0200 fprintf('Take a look at the options that CG used:\n'); 0201 disp(options); 0202 fprintf('And see how many trials were made at each line search call:\n'); 0203 info_ls = [info.linesearch]; 0204 disp([info_ls.costevals]); 0205 0206 0207 0208 0209 0210 0211 0212 0213 fprintf('Try it again without the linesearch helper.\n'); 0214 0215 % Remove the linesearch helper from the problem structure. 0216 problem = rmfield(problem, 'linesearch'); 0217 0218 [Xcg, xcost, info, options] = conjugategradient(problem, X0); %#ok<ASGLU> 0219 0220 fprintf('Take a look at the options that CG used:\n'); 0221 disp(options); 0222 fprintf('And see how many trials were made at each line search call:\n'); 0223 info_ls = [info.linesearch]; 0224 disp([info_ls.costevals]); 0225 0226 0227 0228 0229 0230 % If the problem has a small enough dimension, we may (for analysis 0231 % purposes) compute the spectrum of the Hessian at a point X. This may 0232 % help in studying the conditioning of a problem. If you don't provide 0233 % the Hessian, Manopt will approximate the Hessian with finite 0234 % differences of the gradient and try to estimate its "spectrum" (it's 0235 % not a proper linear operator). This can give some intuition, but 0236 % should not be relied upon. 0237 if exist('OCTAVE_VERSION', 'builtin') == 0 0238 hstgrm = @histogram; 0239 else 0240 hstgrm = @hist; 0241 end 0242 if problem.M.dim() < 2000 0243 fprintf('Computing the spectrum of the Hessian...\n'); 0244 s = hessianspectrum(problem, X); 0245 subplot(1, 2, 1); 0246 hstgrm(s); 0247 title('Histogram of eigenvalues of the Hessian at the solution'); 0248 subplot(1, 2, 2); 0249 hstgrm(log10(abs(s))); 0250 title('Histogram of log_{10}(|eigenvalues|) of the Hessian at the solution'); 0251 end 0252 0253 0254 0255 0256 end

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