# Performance Bounds for Expander-based Compressed Sensing in the presence of Poisson Noise

## TL;DR

This paper develops performance bounds for expander graph-based compressed sensing under Poisson noise, demonstrating their advantages over traditional dense sensing matrices through theoretical analysis and experiments.

## Contribution

It introduces a novel expander graph-based sensing paradigm and a MAP recovery algorithm tailored for Poisson noise, with proven error bounds and improved performance.

## Key findings

- Expander graphs outperform dense matrices in Poisson noise scenarios.
- The proposed MAP algorithm effectively recovers sparse signals.
- Experimental results confirm theoretical bounds and advantages.

## Abstract

This paper provides performance bounds for compressed sensing in the presence of Poisson noise using expander graphs. The Poisson noise model is appropriate for a variety of applications, including low-light imaging and digital streaming, where the signal-independent and/or bounded noise models used in the compressed sensing literature are no longer applicable. In this paper, we develop a novel sensing paradigm based on expander graphs and propose a MAP algorithm for recovering sparse or compressible signals from Poisson observations. The geometry of the expander graphs and the positivity of the corresponding sensing matrices play a crucial role in establishing the bounds on the signal reconstruction error of the proposed algorithm. The geometry of the expander graphs makes them provably superior to random dense sensing matrices, such as Gaussian or partial Fourier ensembles, for the Poisson noise model. We support our results with experimental demonstrations.