The investigators design explicit sensing matrices in concert with signal processing algorithms that outperform the state of the art. They progress towards solutions of several of the most significant open problems in compressed sensing and frame theory: (a) to find deterministic constructions of matrices with the restricted isometry property (RIP); (b) to construct real and complex equiangular tight frames (ETFs), including the maximal ETFs conjectured to exist by Zauner; (c) to design fast and reliable algorithms for phase retrieval;(d) to prove recent refinements of Wright's conjecture regarding the minimal number of injective intensity measurements; (e) the Paulsen problem: How to project onto the manifold of unit norm tight frames (UNTFs).
Threat detection requires high-quality sensing and effective signal processing algorithms. Most modern threats have faint signatures, leaving intelligence agencies with two main problems. First, you can't find what you don't seek. That is, in order to detect a threat, you must first adequately sense the region of interest. Because of this, part of the work focuses on new mathematics that will improve sensing capabilities. This has the unfortunate side-effect of adding to the existing fire hose of data, exacerbating the second main problem: "finding a needle in a haystack." At the moment, analysts must wade through unimaginably large data sets in order to extract actionable intelligence. As such, the remainder of the work focuses on new mathematical ways of processing big data sets.
