A Multiple Hypothesis Testing Approach to Low-Complexity Subspace Unmixing

TL;DR

This paper introduces a low-complexity algorithm called marginal subspace detection (MSD) for identifying active subspaces in high-dimensional data, framing it as a multiple hypothesis testing problem with controlled error rates.

Contribution

It formalizes the subspace unmixing problem under the PS3 model, connects it to various fields, and proposes a scalable, polynomial-time algorithm with theoretical error control.

Findings

MSD effectively controls family-wise error rate at any level α

Applicable to arbitrary collections of subspaces on the Grassmann manifold

Allows linear scaling of active subspaces with ambient dimension

Abstract

Subspace-based signal processing traditionally focuses on problems involving a few subspaces. Recently, a number of problems in different application areas have emerged that involve a significantly larger number of subspaces relative to the ambient dimension. It becomes imperative in such settings to first identify a smaller set of active subspaces that contribute to the observation before further processing can be carried out. This problem of identification of a small set of active subspaces among a huge collection of subspaces from a single (noisy) observation in the ambient space is termed subspace unmixing. This paper formally poses the subspace unmixing problem under the parsimonious subspace-sum (PS3) model, discusses connections of the PS3 model to problems in wireless communications, hyperspectral imaging, high-dimensional statistics and compressed sensing, and proposes a low-complexity algorithm, termed marginal subspace detection (MSD), for subspace unmixing. The MSD algorithm turns the subspace unmixing problem for the PS3 model into a multiple hypothesis testing (MHT) problem and its analysis in the paper helps control the family-wise error rate of this MHT problem at any level $\alpha \in [0,1]$ under two random signal generation models. Some other highlights of the analysis of the MSD algorithm include: (i) it is applicable to an arbitrary collection of subspaces on the Grassmann manifold; (ii) it relies on properties of the collection of subspaces that are computable in polynomial time; and ($iii$) it allows for linear scaling of the number of active subspaces as a function of the ambient dimension. Finally, numerical results are presented in the paper to better understand the performance of the MSD algorithm.