Sparse Signal Recovery under Poisson Statistics

TL;DR

This paper investigates sparse signal recovery under Poisson noise, deriving sample complexity bounds for ML estimators that account for sparsity and parameter scale, with applications in various fields.

Contribution

It introduces the Restricted Likelihood Perturbation (RLP) concept and provides fundamental bounds for Poisson-based sparse recovery, extending beyond Gaussian noise models.

Findings

Sample complexity depends on sparsity and parameter scale.

Derived bounds for both deterministic and random sensing matrices.

Introduced RLP to unify scale and sparsity considerations.

Abstract

We are motivated by problems that arise in a number of applications such as Online Marketing and explosives detection, where the observations are usually modeled using Poisson statistics. We model each observation as a Poisson random variable whose mean is a sparse linear superposition of known patterns. Unlike many conventional problems observations here are not identically distributed since they are associated with different sensing modalities. We analyze the performance of a Maximum Likelihood (ML) decoder, which for our Poisson setting involves a non-linear optimization but yet is computationally tractable. We derive fundamental sample complexity bounds for sparse recovery when the measurements are contaminated with Poisson noise. In contrast to the least-squares linear regression setting with Gaussian noise, we observe that in addition to sparsity, the scale of the parameters also fundamentally impacts sample complexity. We introduce a novel notion of Restricted Likelihood Perturbation (RLP), to jointly account for scale and sparsity. We derive sample complexity bounds for $\ell_1$ regularized ML estimators in terms of RLP and further specialize these results for deterministic and random sensing matrix designs.