  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.
