Compressed Subspace Matching on the Continuum

TL;DR

This paper demonstrates that subspace matching from compressed measurements is nearly as accurate as from full observations, provided the projection dimension scales with the subspace complexity, with applications in template matching and source localization.

Contribution

It introduces a theoretical framework for subspace matching in compressed sensing with continuously parameterized subspaces, establishing conditions for near-optimal recovery.

Findings

Matching quality is preserved under certain measurement dimensions.

The geometrical complexity of subspace collections influences measurement requirements.

Applicable to template matching and source localization problems.

Abstract

We consider the general problem of matching a subspace to a signal in R^N that has been observed indirectly (compressed) through a random projection. We are interested in the case where the collection of K-dimensional subspaces is continuously parameterized, i.e. naturally indexed by an interval from the real line, or more generally a region of R^D. Our main results show that if the dimension of the random projection is on the order of K times a geometrical constant that describes the complexity of the collection, then the match obtained from the compressed observation is nearly as good as one obtained from a full observation of the signal. We give multiple concrete examples of collections of subspaces for which this geometrical constant can be estimated, and discuss the relevance of the results to the general problems of template matching and source localization.