Sparse Signal Recovery under Poisson Statistics
arXiv:1307.4666 · doi:10.1109/Allerton.2013.6736698
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 regularized ML estimators in terms of RLP and further specialize these results for deterministic and random sensing matrix designs.
13 pages, 11 figures, 2 tables, submitted to IEEE Transactions on Signal Processing
References in corpus (8)
- On the conditions used to prove oracle results for the Lasso
- A proximal iteration for deconvolving Poisson noisy images using sparse representations
- Compressed sensing performance bounds under Poisson noise
- Restricted Eigenvalue Conditions on Subgaussian Random Matrices
- Performance bounds for expander-based compressed sensing in Poisson noise
- Learning Exponential Families in High-Dimensions: Strong Convexity and Sparsity
- Sparse Poisson Intensity Reconstruction Algorithms
- Sparse Signal Recovery under Poisson Statistics