Covering point-sets with parallel hyperplanes and sparse signal recovery

Abstract

We give a new deterministic construction of integer sensing matrices that can be used for the recovery of integer-valued signals in compressed sensing. This is a family of $n\times d$ integer matrices, $d\ge n$, with bounded sup-norm and the property that no $\ell$ column vectors are linearly dependent, $\ell \le n$. Further, if $\ell \le o(\log n)$ then $d/n\to \infty$ as $n\to \infty$. Our construction comes from particular sets of difference vectors of point-sets in ${\mathbb{R}}^n$ that cannot be covered by few parallel hyperplanes. We construct examples of such sets on the $0,\pm 1$ grid and use them for the matrix construction. We also show a connection of our constructions to a simple version of the Tarski plank problem.

arXiv comment11 pages, 1 figure; to appear in Discrete and Computational Geometry