Explicit RIP Matrices in Compressed Sensing from Algebraic Geometry

TL;DR

This paper introduces new explicit algebraic geometric constructions of RIP matrices for compressed sensing, achieving smaller coherence and larger sparsity, with some satisfying the strong coherence property for noisy signal recovery.

Contribution

It presents novel algebraic geometric methods to construct RIP matrices with improved properties and extends previous deterministic constructions in compressed sensing.

Findings

RIP matrices from algebraic geometry have smaller coherence and larger sparsity.

The asymptotic bounds of these matrices match previous optimal bounds.

A modified construction satisfies the strong coherence property for noisy measurements.

Abstract

Compressed sensing was proposed by E. J. Cand\'es, J. Romberg, T. Tao, and D. Donoho for efficient sampling of sparse signals in 2006 and has vast applications in signal processing. The expicit restricted isometry property (RIP) measurement matrices are needed in practice. Since 2007 R. DeVore, J. Bourgain et al and R. Calderbank et al have given several deterministic cosntrcutions of RIP matrices from various mathematical objects. On the other hand the strong coherence property of a measurement matrix was introduced by Bajwa and Calderbank et al for the recovery of signals under the noisy measuremnt. In this paper we propose new explicit construction of real valued RIP measurement matrices in compressed sensing from algebraic geometry. Our construction indicates that using more general algebraic-geometric objects rather than curves (AG codes), RIP measurement matrices in compressed sensing can be constructed with much smaller coherence and much bigger sparsity orders. The RIP matrices from algebraic geometry also have a nice asymptotic bound matching the bound from the previous constructions of Bourgain et al and the small-bias sets. On the negative side, we prove that the RIP matrices from DeVore's construction, its direct algebraic geometric generalization and one of our new construction do not satisfy the strong coherence property. However we give a modified version of AG-RIP matrices which satisfies the strong coherence property. Therefore the new RIP matrices in compressed sensing from our modified algebraic geometric construction can be used for the recovery of signals from the noisy measurement.