Wednesday, October 31, 2018 at 11:00am to 11:50am
Kaprielian Hall (KAP), 414
3620 South Vermont Avenue, Los Angeles, CA 90089
Lenny Fukshansky, Claremont McKenna College
We investigate the problem of constructing m by d matrices A with small entries and d large comparing to m so that for all integer vectors x in R^d with at most m nonzero coordinates the Euclidean norm of the image vector Ax is bounded away from 0 by an absolute constant. A bound like this allows for robust recovery of the original sparse vector x from its image Ax. This problem is motivated by the compressed sensing paradigm and has numerous potential applications ranging from wireless communications to medical imaging. We use a combination of combinatorial, probabilistic and number-theoretic methods to discuss existence and some constructions of such sensing matrices with concrete examples. We also discuss limitations of our constructions, stemming from sparse variations of some classical results in the geometry of numbers. This is joint work with D. Needell and B. Sudakov.