The phase transition of matrix recovery from Gaussian measurements matches the minimax MSE of matrix denoising

Let X₀ be an unknown M by N matrix. In matrix recovery, one takes n<MN linear measurements y₁,...,y_n of X ₀, where y_i = Tr(A^T_i X₀) and each Ai is an M by N matrix. A popular approach for matrix recovery is nuclear norm minimization (NNM): solving the convex optimization problem min ||X||_* subject to y_i =Tr(A^T_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 δ(n,M,N)= n/(MN), rank fraction ρ=rank(X₀)/min{M,N}, and aspect ratio β=M/N. Specifically when the measurement matrices A_i have independent standard Gaussian random entries, a curve δ*(ρ)= δ*(ρ;β) exists such that, if δ>δ* (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 min||Y-X||²_F /2+λ||X|| _* . 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,X̂_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.