An asymptotic existence result on compressed sensing matrices

Abstract

MSC: 05B05 42C15 94A08 Keywords: Compressed sensing Pairwise balanced designsFor any rational number h and all sufficiently large n we give a deterministic construction for an n × hn com-Our method uses pairwise balanced designs and complex Hadamard matrices in the construction of -equiangular frames, which we introduce as a generalisation of equiangular tight frames. The method is general and produces good compressed sensing matrices from any appropriately chosen pairwise balanced design. The ( 1 , t)-recoverability performance is specified as a simple function of the parameters of the design. To obtain our asymptotic existence result we prove new results on the existence of pairwise balanced designs in which the numbers of blocks of each size are specified.

Compressed Sensing With Combinatorial Designs: Theory and Simulations

“…Two extremes can be distinguished. When k is constant, v ∼ r ∼ √ n and N ∼ r 2 ∼ n -this is the case considered in [4]. When k ∼ √ v , v ∼ r 2 ∼ n and N ∼ r 3 ∼ n 3 2 (this occurs when D is a projective plane, for example).…”

“…In [4], a construction for compressed sensing matrices based on block designs and complex Hadamard matrices was introduced (see Construction 1 below). Here we add to the analysis of these matrices, both establishing new results on their compressed sensing performance, and providing details of extensive simulations.…”

“…1. Theorem 10 of [4] establishes that an n × N matrix created via Construction 1 has the (ℓ 1 , t)-recovery property for all values of t ≤ √ n 4 , where n is the number of rows in the matrix. In Section 3 we show that there exist vectors of sparsity at most √ 2n which cannot be recovered.…”

“…Algorithm running times are better than for Gaussian matrices. 4. In Section 6 we propose a new algorithm for signal recovery, tailored to Construction 1.…”

“…Because n is the number of blocks in D and Combining Proposition 9 with Theorem 10 of [4] we can specify the (ℓ 1 , t)-recoverability of a matrix obtained from Construction 1 to within a multiplicative factor of 4 √ 2.…”