## 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} = Tr(A^{T}_{i} X_{0}) 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(X_{0})/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_{λ}sup _{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.

Original language | American English |
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Pages (from-to) | 8405-8410 |

Number of pages | 6 |

Journal | Proceedings of the National Academy of Sciences of the United States of America |

Volume | 110 |

Issue number | 21 |

DOIs | |

State | Published - 21 May 2013 |

Externally published | Yes |

## Keywords

- Compressed sensing
- Matrix completion