Limits of Deterministic Compressed Sensing Considering Arbitrary Orthonormal Basis for Sparsity

TL;DR

This paper explores the limitations of deterministic sampling methods in compressed sensing, showing that they generally cannot guarantee unique recovery of sparse signals except in the simplest case, unlike random sampling.

Contribution

It demonstrates that deterministic sampling functions, especially linear ones, are insufficient for unique sparse signal recovery beyond the case of single-sparse signals.

Findings

Deterministic linear samples fail for k-sparse signals with k > 1.

Deterministic nonlinear functions can uniquely recover 1-sparse signals.

Random sampling guarantees are not replicable with deterministic methods.

Abstract

It is previously shown that proper random linear samples of a finite discrete signal (vector) which has a sparse representation in an orthonormal basis make it possible (with probability 1) to recover the original signal. Moreover, the choice of the linear samples does not depend on the sparsity domain. In this paper, we will show that the replacement of random linear samples with deterministic functions of the signal (not necessarily linear) will not result in unique reconstruction of k-sparse signals except for k=1. We will show that there exist deterministic nonlinear sampling functions for unique reconstruction of 1- sparse signals while deterministic linear samples fail to do so.