False Discovery and Its Control in Low Rank Estimation

TL;DR

This paper introduces a geometric framework for assessing true and false discoveries in low-rank matrix estimation, extending stability selection methods to control false discoveries in this context.

Contribution

It develops a novel geometric reformulation of discovery concepts and generalizes stability selection to effectively control false discoveries in low-rank estimation.

Findings

The proposed method effectively controls false discoveries in low-rank estimation.

Numerical experiments demonstrate improved performance over previous approaches.

The framework provides a formal basis for inference on low-rank structures.

Abstract

Models specified by low-rank matrices are ubiquitous in contemporary applications. In many of these problem domains, the row/column space structure of a low-rank matrix carries information about some underlying phenomenon, and it is of interest in inferential settings to evaluate the extent to which the row/column spaces of an estimated low-rank matrix signify discoveries about the phenomenon. However, in contrast to variable selection, we lack a formal framework to assess true/false discoveries in low-rank estimation; in particular, the key source of difficulty is that the standard notion of a discovery is a discrete one that is ill-suited to the smooth structure underlying low-rank matrices. We address this challenge via a geometric reformulation of the concept of a discovery, which then enables a natural definition in the low-rank case. We describe and analyze a generalization of the Stability Selection method of Meinshausen and B\"uhlmann to control for false discoveries in low-rank estimation, and we demonstrate its utility compared to previous approaches via numerical experiments.