Paper detail

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

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 $y_i = \Tr(a_i^T X_0)$ 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). Empirical work reveals a \emph{phase transition} curve, stated in terms of the undersampling fraction $δ(n,M,N) = n/(MN)$, rank fraction $ρ=r/N$ and aspect ratio $β=M/N$. Specifically, a curve $δ^* = δ^*(ρ;β)$ exists such that, if $δ> δ^*(ρ;β)$, NNM typically succeeds, while if $δ< δ^*(ρ;β)$, 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 $\text{min} \| Y - X \|_F^2/2 + λ\|X\|_* $. When optimally tuned, this scheme achieves the asymptotic minimax MSE $\cM(ρ) = \lim_{N \goto \infty} \inf_λ\sup_{\rank(X) \leq ρ\cdot N} MSE(X,\hat{X}_λ)$. We report extensive experiments showing that the phase transition $δ^*(ρ)$ in the first problem coincides with the minimax risk curve $\cM(ρ)$ in the second problem, for {\em any} rank fraction $0 < ρ< 1$.