The Phase Transition of Matrix Recovery from Gaussian Measurements Matches the Minimax MSE of Matrix Denoising
By David Donoho, Matan Gavish, and Andrea Montanari
Stanford University (2013)
Abstract Paper Matan  Gavish

The program provided calculates the asymptotic minimax MSE, and the asymptotic minimax tuning threshold, of matrix denoising by Singular Value Thresholding: lim_{N->\infty} inf_lambda sup_{rank(X)<= M*rho} MSE ||Xhat_lambda - X||^2_F /MN ... Here: (*) Xhat_lambda is the Singular Value Thresholding denoiser (applying soft thresholding with threshold lambda to each singular value of the data) (*) rho is the asymptotic rank fraction (*) M/N -> beta (the asymptotic aspect ratio) (*) X is an M-by-N matrix, M<=N (*) ||.||_F denotes the Frobenius matrix norm (sum of squares of matrix entries) Created
February 09, 2013 Software:
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Abstract
Let X_0 be an unknown M by N matrix. In matrix recovery, one takes n < MN linear measurements y_1, ... , y_n of X_0, where y_i = Trace(a_i' X_0) and each a_i is a M by N matrix. For measurement matrices with Gaussian i.i.d entries, it known that if X_0 is of low rank, it is recoverable from just a few measurements. A popular approach for matrix recovery is Nuclear Norm Minimization (NNM): solving the convex optimization problem min ||X||_* subject to y_i=Trace(a_i' X) for all 1<= i<= n, where || . ||_* denotes the nuclear norm, namely, the sum of singular values. Empirical work reveals a phase transition curve, stated in terms of the undersampling fraction \delta(n,M,N) = n/(MN), rank fraction \rho=r/N and aspect ratio \beta=M/N. Specifically, a curve \delta^* = \delta^*(\rho;\beta) exists such that, if \delta > \delta^*(\rho;\beta), NNM typically succeeds, while if \delta < \delta^*(\rho;\beta), it typically fails. An apparently quite different problem is matrix denoising in Gaussian noise, where an unknown M by N matrix X_0 is to be estimated based on direct noisy measurements Y = X_0 + Z, where the matrix Z has iid Gaussian entries. It has been empirically observed that, if X_0 has low rank, it may be recovered quite accurately from the noisy measurement Y. A popular matrix denoising scheme solves the unconstrained optimization problem min || Y - X ||_F^2/2 + \lambda ||X||_*. When optimally tuned, this scheme achieves the asymptotic minimax MSE, M( \rho ) = \lim_{N-> \infty} \inf_\lambda \sup_{\rank(X) <= \rho * N} MSE(X,\hat{X}_\lambda). We report extensive experiments showing that the phase transition \delta^*(\rho) in the first problem (Matrix Recovery from Gaussian Measurements) coincides with the minimax risk curve M(\rho) in the second problem (Matrix Denoising in Gaussian Noise): \delta^*(\rho) = M(\rho), for any rank fraction 0 < \rho < 1. Our experiments considered matrices belonging to two constraint classes: real M by N matrices, of various ranks and aspect ratios, and real symmetric positive semidefinite N by N matrices, of various ranks. Different predictions M(\rho) of the phase transition location were used in the two different cases, and were validated by the experimental data.
Donoho, D., M. Gavish, and A. Montanari, "The Phase Transition of Matrix Recovery from Gaussian Measurements Matches the Minimax MSE of Matrix Denoising", Stanford University.
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β Waiting time Please cite the publication as :

Donoho, D., M. Gavish, and A. Montanari, "The Phase Transition of Matrix Recovery from Gaussian Measurements Matches the Minimax MSE of Matrix Denoising", Stanford University.

Please cite the companion website as :

Donoho, D., M. Gavish, and A. Montanari, "The Phase Transition of Matrix Recovery from Gaussian Measurements Matches the Minimax MSE of Matrix Denoising", RunMyCode companion website, http://www.execandshare.org/CompanionSite/Site265 Case
'mat' for general matrices, 'sym' for symmetric positive-semidefinite matrices
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rank fraction, r/M for an M-by-N matrix
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aspect ratio, M/N for M-by-N matrix Variable/Parameters Description Visualisation
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β The Phase Transition of Matrix Recovery from Gaussian Measurements Matches the Minimax MSE of Matrix Denoising
M. Gavish (2013) Computing Date Status Actions
Coder:   • Matan Gavish

Matan Gavish also created these companion sites

 The Optimal Hard Threshold for Singular Values is 4/sqrt(3) Abstract We consider recovery of low-rank matrices from noisy data by hard thresholding of singular values, where singular values below a prescribed threshold \lambda are set to 0. We study the asymptotic MSE in a framework where the matrix size is large compared to the rank of the matrix to be recovered, and the signal-to-noise ratio of the low-rank piece stays constant. The AMSE-optimal choice of hard threshold, in the case of n-by-n matrix in noise level \sigma, is simply (4/\sqrt{3}) \sqrt{n}\sigma \approx 2.309 \sqrt{n}\sigma when \sigma is known, or simply 2.858\cdot y_{med} when \sigma is unknown, where y_{med} is the median empirical singular value. For nonsquare m by n matrices with m \neq n, these thresholding coefficients are replaced with different provided constants. In our asymptotic framework, this thresholding rule adapts to unknown rank and to unknown noise level in an optimal manner: it is always better than hard thresholding at any other value, no matter what the matrix is that we are trying to recover, and is always better than ideal Truncated SVD (TSVD), which truncates at the true rank of the low-rank matrix we are trying to recover. Hard thresholding at the recommended value to recover an n-by-n matrix of rank r guarantees an AMSE at most 3nr\sigma^2. In comparison, the guarantee provided by TSVD is 5nr\sigma^2, the guarantee provided by optimally tuned singular value soft thresholding is 6nr\sigma^2, and the best guarantee achievable by any shrinkage of the data singular values is 2nr\sigma^2. Empirical evidence shows that these AMSE properties of the 4/\sqrt{3} thresholding rule remain valid even for relatively small n, and that performance improvement over TSVD and other shrinkage rules is substantial, turning it into the practical hard threshold of choice. Gavish, M., and D. Donoho, "The Optimal Hard Threshold for Singular Values is 4/sqrt(3)", Stanford University. Authors: Donoho Gavish Coders: Gavish Donoho Last update 05/30/2013 Ranking 9999 Runs 17 Visits N.A.

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The Phase Transition of Matrix Recovery from Gaussian Measurements Matches the Minimax MSE of Matrix Denoising

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 Sparse Nonnegative Solution of Underdetermined Linear Equations by Linear Programming Abstract Consider an underdetermined system of linear equations y = Ax with known d*n matrix A and known y. We seek the sparsest nonnegative solution, i.e. the nonnegative x with fewest nonzeros satisfying y = Ax. In general this problem is NP-hard. However, for many matrices A there is a threshold phenomenon: if the sparsest solution is sufficiently sparse, it can be found by linear programming. In classical convex polytope theory, a polytope P is called k-neighborly if every set of k vertices of P span a face of P. Let aj denote the j-th column of A, 1<=_j<=_n, let a0 = 0 and let P denote the convex hull of the aj . We say P is outwardly k-neighborly if every subset of k vertices not including 0 spans a face of P. We show that outward k-neighborliness is completely equivalent to the statement that, whenever y = Ax has a nonnegative solution with at most k nonzeros, it is the nonnegative solution to y = Ax having minimal sum. Using this and classical results on polytope neighborliness we obtain two types of corollaries. First, because many [d/2]-neighborly polytopes are known, there are many systems where the sparsest solution is available by convex optimization rather than combinatorial optimization - provided the answer has fewer nonzeros than half the number of equations. We mention examples involving incompletely-observed Fourier transforms and Laplace transforms. Second, results on classical neighborliness of high-dimensional randomly-projected simplices imply that, if A is a typical uniformly-distributed random orthoprojector with n = 2d and n large, the sparsest nonnegative solution to y = Ax can be found by linear programming provided it has fewer nonzeros than 1/8 the number of equations. We also consider a notion of weak neighborliness, in which the overwhelming majority of k-sets of aj's not containing 0 span a face. This implies that most nonnegative vectors x with k nonzeros are uniquely determined by y = Ax. As a corollary of recent work counting faces of random simplices, it is known that most polytopes P generated by large n by 2n uniformly-distributed orthoprojectors A are weakly k-neighborly with k _~.558n. We infer that for most n by 2n underdetermined systems having a sparse solution with fewer nonzeros than roughly half the number of equations, the sparsest solution can be found by linear programming. Donoho, D., and J. Tanner, "Sparse Nonnegative Solution of Underdetermined Linear Equations by Linear Programming ", Proceedings of the National Academy of Sciences of the United States of America, 102, 9446–9451. Authors: Donoho Tanner Coders: Donoho Tanner Last update 10/08/2012 Ranking 17 Runs 6 Visits 81 Neighborliness of Randomly-Projected Simplices in High Dimensions Abstract Let A be a d by n matrix, d < n. Let T = T^n-1 be the standard regular simplex in R^n. We count the faces of the projected simplex AT in the case where the projection is random, the dimension d is large and n and d are comparable: d ~ dn, d in (0, 1). The projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of R^n. We derive ?N( d) > 0 with the property that, for any ? < ? N( deter), with overwhelming probability for large d, the number of k-dimensional faces of P = AT is exactly the same as for T, for 0<=k<= ?d. This implies that P is [?d]-neighborly, and its skeleton Skel[? d] ( P) is combinatorially equivalent to Skel[?d] (T). We display graphs of ?N. We also study a weaker notion of neighborliness it asks if the k-faces are all simplicial and if the numbers of k-dimensional faces fk(P) >= fk(T)(1-e). This was already considered by Vershik and Sporyshev, who obtained qualitative results about the existence of a threshold ? VS(d) > 0 at which phase transition occurs in k/d. We compute and display ?VS and compare to ?N. Our results imply that the convex hull of n Gaussian samples in R^d, with n large and proportional to d, ‘looks like a simplex’ in the following sense. In a typical realization of such a high-dimensional Gaussian point cloud d~ dn, all points are on the boundary of the convex hull, and all pairwise line segments, triangles, quadrangles, …, [?d]-angles are on the boundary, for ? 0 with the property that, for any ? < ?N(d), with overwhelming probability for large d, the number of k-dimensional faces of P = AC is the same as for C, for 0 k d. This implies that P is centrally bdc-neighborly, and its skeleton Skel[? d](P) is combinatorially equivalent to Skel[? d]©. We display graphs of ?N. Two weaker notions of neighborliness are also important for understanding sparse solutions of linear equations: facial neighborliness and sectional neighborliness ; we study both. The weakest, (k, e)-facial neighborliness, asks if the k-faces are all simplicial and if the numbers of k-dimensional faces fk(P) >= fk(C)(1 - e). We characterize and compute the critical proportion ?F (d) > 0 at which phase transition occurs in k/d. The other, (k, e)- sectional neighborliness, asks whether all, except for a small fraction epsilon, of the k-dimensional intrinsic sections of P are k-dimensional cross-polytopes. (Intrinsic sections intersect P with k-dimensional subspaces spanned by vertices of P.) We characterize and compute a proportion ?S(d) > 0 guaranteeing this property for k/d ~ ? < ?S(d). We display graphs of ?S and ?F. Donoho, D., "High-Dimensional Centrally-Symmetric Polytopes With Neighborliness Proportional to Dimension", Discrete & Computational Geometry, 35, 617-652. Authors: Donoho Coders: Donoho Last update 10/08/2012 Ranking 16 Runs 3 Visits 33 The Optimal Hard Threshold for Singular Values is 4/sqrt(3) Abstract We consider recovery of low-rank matrices from noisy data by hard thresholding of singular values, where singular values below a prescribed threshold \lambda are set to 0. We study the asymptotic MSE in a framework where the matrix size is large compared to the rank of the matrix to be recovered, and the signal-to-noise ratio of the low-rank piece stays constant. The AMSE-optimal choice of hard threshold, in the case of n-by-n matrix in noise level \sigma, is simply (4/\sqrt{3}) \sqrt{n}\sigma \approx 2.309 \sqrt{n}\sigma when \sigma is known, or simply 2.858\cdot y_{med} when \sigma is unknown, where y_{med} is the median empirical singular value. For nonsquare m by n matrices with m \neq n, these thresholding coefficients are replaced with different provided constants. In our asymptotic framework, this thresholding rule adapts to unknown rank and to unknown noise level in an optimal manner: it is always better than hard thresholding at any other value, no matter what the matrix is that we are trying to recover, and is always better than ideal Truncated SVD (TSVD), which truncates at the true rank of the low-rank matrix we are trying to recover. 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