Matrix Completion with Nonconvex Regularization: Spectral Operators and Scalable Algorithms

Mazumder, Rahul; Saldana, Diego Franco; Weng, Haolei

doi:10.48550/arxiv.1801.08227

Cited by 1 publication

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“…See Section 5 for the explicit expressions. Nonconvex penalties are well known to attenuate the estimation bias caused by convex sparsity promotion functions [10,21,23]. Note that some popular non-convex penalties like P θ (α) = θ|α| q (0 ≤ q < 1)…”

Section: Estimators Obtained Via Spectral Regularizationmentioning

confidence: 99%

Computing the degrees of freedom of rank-regularized estimators and cousins

Mazumder¹,

Weng²

2020

Electron. J. Statist.

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Estimating a low rank matrix from its linear measurements is a problem of central importance in contemporary statistical analysis. The choice of tuning parameters for estimators remains an important challenge from a theoretical and practical perspective. To this end, Stein's Unbiased Risk Estimate (SURE) framework provides a well-grounded statistical framework for degrees of freedom estimation. In this paper, we use the SURE framework to obtain degrees of freedom estimates for a general class of spectral regularized matrix estimators, generalizing beyond the class of estimators that have been studied thus far. To this end, we use a result due to Shapiro (2002) pertaining to the differentiability of symmetric matrix valued functions, developed in the context of semidefinite optimization algorithms. We rigorously verify the applicability of Stein's lemma towards the derivation of degrees of freedom estimates; and also present new techniques based on Gaussian convolution to estimate the degrees of freedom of a class of spectral estimators to which Stein's lemma is not directly applicable.

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Section: Estimators Obtained Via Spectral Regularizationmentioning

confidence: 99%