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May 2018 Flexible Low-Rank Statistical Modeling with Missing Data and Side Information
William Fithian, Rahul Mazumder
Statist. Sci. 33(2): 238-260 (May 2018). DOI: 10.1214/18-STS642

Abstract

We explore a general statistical framework for low-rank modeling of matrix-valued data, based on convex optimization with a generalized nuclear norm penalty. We study several related problems: the usual low-rank matrix completion problem with flexible loss functions arising from generalized linear models; reduced-rank regression and multi-task learning; and generalizations of both problems where side information about rows and columns is available, in the form of features or smoothing kernels. We show that our approach encompasses maximum a posteriori estimation arising from Bayesian hierarchical modeling with latent factors, and discuss ramifications of the missing-data mechanism in the context of matrix completion. While the above problems can be naturally posed as rank-constrained optimization problems, which are nonconvex and computationally difficult, we show how to relax them via generalized nuclear norm regularization to obtain convex optimization problems. We discuss algorithms drawing inspiration from modern convex optimization methods to address these large scale convex optimization computational tasks. Finally, we illustrate our flexible approach in problems arising in functional data reconstruction and ecological species distribution modeling.

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William Fithian. Rahul Mazumder. "Flexible Low-Rank Statistical Modeling with Missing Data and Side Information." Statist. Sci. 33 (2) 238 - 260, May 2018. https://doi.org/10.1214/18-STS642

Information

Published: May 2018
First available in Project Euclid: 3 May 2018

zbMATH: 1397.62180
MathSciNet: MR3797712
Digital Object Identifier: 10.1214/18-STS642

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.33 • No. 2 • May 2018
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