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

arXiv: https://arxiv.org/abs/1909.10143 · HTML: https://mlsanity.com/paper/1909.10143/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Sep 2019.

## Built on by (most cited first, 2 of 2)

- [Bayesian Inference of Psychometric Variables From Brain and Behavior in Implicit Association Tests](https://mlsanity.com/paper/2603.16741/) (2026)
- [When Does Dense Retrieval Need Asymmetric Geometry? A Bias-Variance Theory of Shared and Dual Projections](https://mlsanity.com/paper/2609.32488/) (2026)

## Abstract (arXiv)

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.