Minimax Optimal Sparse Signal Recovery with Poisson Statistics

TL;DR

This paper establishes fundamental limits and optimal methods for recovering sparse signals from Poisson-distributed observations, which are relevant in applications like marketing and detection, highlighting the impact of scale and sparsity.

Contribution

The paper derives sample complexity bounds and proves the minimax optimality of a constrained ML decoder for sparse Poisson signal recovery.

Findings

Sample complexity bounds for Poisson sparse recovery

The scale of parameters affects error bounds

The proposed ML decoder is minimax optimal

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 $\ell_2$ error in the Poisson setting. We show tightness of our upper bounds both theoretically and experimentally. In particular, we derive a minimax matching lower bound on the mean-squared error and show that our constrained ML decoder is minimax optimal for this regime.