A Multiple Hypothesis Testing Approach to Low-Complexity Subspace Unmixing

Subspace-based signal processing has a rich history in the literature. Traditional focus in this direction has been on problems involving a few subspaces. But a number of problems in different application areas have emerged in recent years that involve 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 observations before further information processing tasks can be carried out. We term 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 as subspace unmixing. In this paper, we formally pose the subspace unmixing problem, discuss its connections with problems in wireless communications, hyperspectral imaging, high-dimensional statistics and compressed sensing, and propose and analyze a low-complexity algorithm, termed marginal subspace detection (MSD), for subspace unmixing. The MSD algorithm turns the subspace unmixing problem into a multiple hypothesis testing (MHT) problem and our analysis helps control the family-wise error rate of this MHT problem at any level. Some other highlights of our 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, we also present numerical results in the paper to better understand the performance of the MSD algorithm.