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 $y1,..., yn$ of $X0$, where $yi = \Tr(ai^T X0)$ and each $ai$ is a $M$ by $N$ matrix. For measurement matrices with Gaussian i.i.d entries, it known that if $X0$ is of low rank, it is recoverable from just a few measurements. A popular approach for matrix recovery is Nuclear Norm Minimization (NNM). Empirical work reveals a \emph{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 $X0$ is to be estimated based on direct noisy measurements $Y = X0 + Z$, where the matrix $Z$ has iid Gaussian entries. It has been empirically observed that, if $X0$ has low rank, it may be recovered quite accurately from the noisy measurement $Y$. A popular matrix denoising scheme solves the unconstrained optimization problem $\text{min} | Y - X |F^2/2 + \lambda |X|* $. When optimally tuned, this scheme achieves the asymptotic minimax MSE $\cM(\rho) = \lim{N \goto \infty} \inf\lambda \sup{\rank(X) \leq \rho \cdot N} MSE(X,\hat{X}\lambda)$. We report extensive experiments showing that the phase transition $\delta^*(\rho)$ in the first problem coincides with the minimax risk curve $\cM(\rho)$ in the second problem, for {\em any} rank fraction $0 < \rho < 1$.