Minimax risk of matrix denoising by singular value thresholding. (English) Zbl 1310.62014

Summary: 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 (\(r_{n}/m_{n}\to \rho\), \(m_{n}/n\to \beta\)), we evaluate the asymptotic minimax MSE \( \mathcal{M} (\rho,\beta)=\lim_{m_{n},n\to \infty}\inf_{\lambda}\sup_{\operatorname{rank}(X)\leq r_{n}}\operatorname{MSE}(X_{0},\hat{X}_{\lambda})\). 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.


62C20 Minimax procedures in statistical decision theory
62H25 Factor analysis and principal components; correspondence analysis
90C25 Convex programming
90C22 Semidefinite programming


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