The Annals of Statistics

Minimax risk of matrix denoising by singular value thresholding

David Donoho and Matan Gavish

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An unknown $m$ by $n$ matrix $X_{0}$ is to be estimated from noisy measurements $Y=X_{0}+Z$, where the noise matrix $Z$ has i.i.d. Gaussian entries. A popular matrix denoising scheme solves the nuclear norm penalization problem $\operatorname{min}_{X}\|Y-X\|_{F}^{2}/2+\lambda\|X\|_{*}$, where $\|X\|_{*}$ denotes the nuclear norm (sum of singular values). This is the analog, for matrices, of $\ell_{1}$ penalization in the vector case. It has been empirically observed that if $X_{0}$ has low rank, it may be recovered quite accurately from the noisy measurement $Y$.

In a proportional growth framework where the rank $r_{n}$, number of rows $m_{n}$ and number of columns $n$ all tend to $\infty$ proportionally to each other ($r_{n}/m_{n}\rightarrow \rho$, $m_{n}/n\rightarrow \beta$), we evaluate the asymptotic minimax MSE $ \mathcal{M} (\rho,\beta)=\lim_{m_{n},n\rightarrow \infty}\inf_{\lambda}\sup_{\operatorname{rank}(X)\leq r_{n}}\operatorname{MSE}(X_{0},\hat{X}_{\lambda})$. Our formulas involve incomplete moments of the quarter- and semi-circle laws ($\beta=1$, square case) and the Marčenko–Pastur law ($\beta<1$, nonsquare case). For finite $m$ and $n$, we show that MSE increases as the nonzero singular values of $X_{0}$ grow larger. As a result, the finite-$n$ worst-case MSE, a quantity which can be evaluated numerically, is achieved when the signal $X_{0}$ is “infinitely strong.”

The nuclear norm penalization problem is solved by applying soft thresholding to the singular values of $Y$. We also derive the minimax threshold, namely the value $\lambda^{*}(\rho)$, which is the optimal place to threshold the singular values.

All these results are obtained for general (nonsquare, nonsymmetric) real matrices. Comparable results are obtained for square symmetric nonnegative-definite matrices.

Article information

Ann. Statist., Volume 42, Number 6 (2014), 2413-2440.

First available in Project Euclid: 20 October 2014

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C20: Minimax procedures 62H25: Factor analysis and principal components; correspondence analysis
Secondary: 90C25: Convex programming 90C22: Semidefinite programming

Matrix denoising nuclear norm minimization singular value thresholding optimal threshold Stein unbiased risk estimate monotonicity of power functions of multivariate tests matrix completion from Gaussian measurements phase transition


Donoho, David; Gavish, Matan. Minimax risk of matrix denoising by singular value thresholding. Ann. Statist. 42 (2014), no. 6, 2413--2440. doi:10.1214/14-AOS1257.

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Supplemental materials

  • Supplementary material: Proofs and additional discussion. In this supplementary material we prove Theorems 5, 6, 7, 8 and other lemmas. We also derive the Stein unbiased risk Estimate (SURE) for SVST, which is instrumental in the proof of Theorem 1. Finally, we discuss similarities between singular value thresholding and sparse vector thresholding.