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

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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)
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February 09, 2013
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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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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

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Case
    'mat' for general matrices, 'sym' for symmetric positive-semidefinite matrices
    ρ
      rank fraction, r/M for an M-by-N matrix
      β
        aspect ratio, M/N for M-by-N matrix
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        The Phase Transition of Matrix Recovery from Gaussian Measurements Matches the Minimax MSE of Matrix Denoising
        M. Gavish (2013)
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