On unique recovery of finite-valued integer signals and admissible lattices of sparse hypercubes

Abstract

The paper considers the problem of unique recovery of sparse finite-valued integer signals using a single linear integer measurement. For l-sparse signals in {mathbb {Z}}^n, 2l<n, with absolute entries bounded by r, we construct an 1times n measurement matrix with maximum absolute entry Delta =O(r^{2l-1}). Here the implicit constant depends on l and n and the exponent 2l-1 is optimal. Additionally, we show that, in the above setting, a single measurement can be replaced by several measurements with absolute entries sub-linear in Delta. The proofs make use of results on admissible (n-1)-dimensional integer lattices for m-sparse n-cubes that are of independent interest.