On the Doubly Sparse Compressed Sensing Problem

TL;DR

This paper introduces a new variant of compressed sensing that tolerates a bounded number of corrupted measurements, providing theoretical bounds, a recovery algorithm, and demonstrating the optimality of measurement matrices.

Contribution

It proposes a novel sparse sensing problem with bounded errors, offers a simple recovery algorithm, and establishes optimal measurement bounds analogous to coding theory.

Findings

Recovery with 2(t+l) measurements is sufficient.

A simple recovery algorithm is proposed.

Measurement matrices are proven optimal via an analog of the Singleton bound.

Abstract

A new variant of the Compressed Sensing problem is investigated when the number of measurements corrupted by errors is upper bounded by some value l but there are no more restrictions on errors. We prove that in this case it is enough to make 2(t+l) measurements, where t is the sparsity of original data. Moreover for this case a rather simple recovery algorithm is proposed. An analog of the Singleton bound from coding theory is derived what proves optimality of the corresponding measurement matrices.