The Annals of Statistics

Minimax risk of matrix denoising by singular value thresholding

David Donoho and Matan Gavish

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Abstract

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

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

Dates
First available in Project Euclid: 20 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aos/1413810732

Digital Object Identifier
doi:10.1214/14-AOS1257

Mathematical Reviews number (MathSciNet)
MR3269984

Zentralblatt MATH identifier
1310.62014

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

Keywords
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

Citation

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. https://projecteuclid.org/euclid.aos/1413810732


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References

  • [1] Anderson, T. W. (1955). The integral of a symmetric unimodal function over a symmetric convex set and some probability inequalities. Proc. Amer. Math. Soc. 6 170–176.
  • [2] Candès, E. J., Sing-Long, C. A. and Trzasko, J. D. (2013). Unbiased risk estimates for singular value thresholding and spectral estimators. IEEE Trans. Signal Process. 61 4643–4657.
  • [3] Das Gupta, S., Anderson, T. W. and Mudholkar, G. S. (1964). Monotonicity of the power functions of some tests of the multivariate linear hypothesis. Ann. Math. Statist. 35 200–205.
  • [4] Donoho, D. L. and Gavish, M. (2013). Companion website for the article the phase transition of matrix recovery from Gaussian measurements matches the minimax MSE of matrix denoising. Available at http://www.runmycode.org/CompanionSite/Site265.
  • [5] Donoho, D. and Gavish, M. (2014). Supplement to “Minimax risk of matrix denoising by singular value thresholding.” DOI:10.1214/14-AOS1257SUPP.
  • [6] Donoho, D. L., Gavish, M. and Montanari, A. (2013). The phase transition of matrix recovery from Gaussian measurements matches the minimax MSE of matrix denoising. Proc. Natl. Acad. Sci. USA 110 8405–8410.
  • [7] Donoho, D. L., Johnstone, I. and Montanari, A. (2013). Accurate prediction of phase transitions in compressed sensing via a connection to minimax denoising. IEEE Trans. Inform. Theory 59 3396–3433.
  • [8] Donoho, D. L. and Johnstone, I. M. (1994). Minimax risk overl $\ell_p$-balls forl $\ell_q$-error. Probab. Theory Related Fields 303 277–303.
  • [9] Donoho, D. L., Johnstone, I. M., Hoch, J. C. and Stern, A. S. (1992). Maximum entropy and the nearly black object. J. R. Stat. Soc. Ser. B Stat. Methodol. 54 41–81.
  • [10] Gavish, M. and Donoho, D. L. (2014). The optimal hard threshold for singular values is $4/\sqrt3$. IEEE Trans. Inform. Theory 60 5040–5053.
  • [11] Grant, M. and Boyd, S. P. (2010). CVX: Matlab software for disciplined convex programming, version 2.0 beta. Available at http://cvxr.com/cvx, September 2013.
  • [12] Koltchinskii, V., Lounici, K. and Tsybakov, A. B. (2011). Nuclear-norm penalization and optimal rates for noisy low-rank matrix completion. Ann. Statist. 39 2302–2329.
  • [13] Lewis, A. S. (1995). The convex analysis of unitarily invariant matrix functions. J. Convex Anal. 2 173–183.
  • [14] Lewis, A. S. and Sendov, H. S. (2005). Nonsmooth analysis of singular values. I. Theory. Set-Valued Var. Anal. 13 213–241.
  • [15] Marcenko, V. and Pastur, L. (1967). Distribution of eigenvalues for some sets of random matrices. Mathematics USSR Sbornik 1 457–483.
  • [16] Oymak, S. and Hassibi, B. (2010). New null space results and recovery thresholds for matrix rank minimization. Preprint. Available at http://arxiv.org/pdf/1011.6326v1.pdf.
  • [17] Oymak, S. and Hassibi, B. (2012). On a relation between the minimax risk and the phase transitions of compressed recovery. In 2012 50th Annual Allerton Conference on Communication, Control, and Computing 1018–1025. IEEE, Piscataway, NJ. Available at http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=6483330.
  • [18] Recht, B., Fazel, M. and Parrilo, P. A. (2010). Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization. SIAM Rev. 52 471–501.
  • [19] Recht, B., Xu, W. and Hassibi, B. (2008). Necessary and sufficient conditions for success of the nuclear norm heuristic for rank minimization. In Proceedings of the 47th IEEE Conference on Decision and Control Cancun, Mexico.
  • [20] Recht, B., Xu, W. and Hassibi, B. (2011). Null space conditions and thresholds for rank minimization. Math. Program. 127 175–202.
  • [21] Rohde, A. and Tsybakov, A. B. (2011). Estimation of high-dimensional low-rank matrices. Ann. Statist. 39 887–930.
  • [22] Shabalin, A. and Nobel, A. (2010). Reconstruction of a low-rank matrix in the presence of Gaussian noise. Preprint. Available at arXiv:1007.4148.
  • [23] Stein, C. M. (1981). Estimation of the mean of a multivariate normal distribution. Ann. Statist. 9 1135–1151.
  • [24] Tanner, J. and Wei, K. (2013). Normalized iterative hard thresholding for matrix completion. SIAM J. Sci. Comput. 35 S104–S125.
  • [25] Zanella, A., Chiani, M. and Win, M. Z. (2009). On the marginal distribution of the eigenvalues of Wishart matrices. IEEE Transactions on Communications 57 1050–1060.

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.