Let X_0 be an unknown M by N matrix. In matrix recovery, one takes nδ* (ρ), NNM typically succeeds for large M,N, whereas if δ<δ *(ρ), it typically fails. An apparently quite different problem is matrix denoising in Gaussian noise, in which 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 independent and identically distributed Gaussian entries. A popular matrix denoising scheme solves the unconstrained optimization problem minY-X~2 _F /2+λX _* . When optimally tuned, this scheme achieves the asymptotic minimax mean-squared error M(ρ;β)= lim _(M,N→∞)inf_λsup _(rank(X)≤ρ·M)MSE(X,X?_λ), where M/N→β. We report extensive experiments showing that the phase transition δ*(ρ) in the first problem, matrix recovery from Gaussian measurements, coincides with the minimax risk curveM(ρ)= M(ρ;β) in the second problem, matrix denoising in Gaussian noise: δ*(ρ)=M(ρ), for any rank fraction 0<ρ<1 (at each common aspect ratio β). 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.
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