Open Access
2020 Computing the degrees of freedom of rank-regularized estimators and cousins
Rahul Mazumder, Haolei Weng
Electron. J. Statist. 14(1): 1348-1385 (2020). DOI: 10.1214/20-EJS1681


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—our results generalize 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, for which Stein’s Lemma does not directly apply.


Download Citation

Rahul Mazumder. Haolei Weng. "Computing the degrees of freedom of rank-regularized estimators and cousins." Electron. J. Statist. 14 (1) 1348 - 1385, 2020.


Received: 1 September 2019; Published: 2020
First available in Project Euclid: 27 March 2020

zbMATH: 07200231
MathSciNet: MR4079794
Digital Object Identifier: 10.1214/20-EJS1681

Primary: 62H12

Keywords: Degrees of freedom , divergence , low rank , matrix valued function , regularization , spectral function , SURE

Vol.14 • No. 1 • 2020
Back to Top