Matrix denoising with partial noise statistics: optimal singular value shrinkage of spiked F-matrices

Author:

Gavish Matan1,Leeb William2,Romanov Elad3

Affiliation:

1. School of Computer Science and Engineering, Hebrew University of Jerusalem , Jerusalem , Israel

2. School of Mathematics, University of Minnesota , Minneapolis, MN , USA

3. Department of Statistics, Stanford University , Stanford, CA , USA

Abstract

Abstract We study the problem of estimating a large, low-rank matrix corrupted by additive noise of unknown covariance, assuming one has access to additional side information in the form of noise-only measurements. We study the Whiten-Shrink-reColour (WSC) workflow, where a ‘noise covariance whitening’ transformation is applied to the observations, followed by appropriate singular value shrinkage and a ‘noise covariance re-colouring’ transformation. We show that under the mean square error loss, a unique, asymptotically optimal shrinkage nonlinearity exists for the WSC denoising workflow, and calculate it in closed form. To this end, we calculate the asymptotic eigenvector rotation of the random spiked F-matrix ensemble, a result which may be of independent interest. With sufficiently many pure-noise measurements, our optimally tuned WSC denoising workflow outperforms, in mean square error, matrix denoising algorithms based on optimal singular value shrinkage that do not make similar use of noise-only side information; numerical experiments show that our procedure’s relative performance is particularly strong in challenging statistical settings with high dimensionality and large degree of heteroscedasticity.

Funder

Hebrew University of Jerusalem Einstein-Kaye scholarship

Israel Science Foundation

NSF BIGDATA

BSF

NSF CAREER

Publisher

Oxford University Press (OUP)

Subject

Applied Mathematics,Computational Theory and Mathematics,Numerical Analysis,Statistics and Probability,Analysis

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