## 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.

### MSC:

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

CVX
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### References:

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