Channel Simulation and Distributed Compression with Ensemble Rejection Sampling

We study channel simulation and distributed matching, two fundamental problems with several applications to machine learning, using a recently introduced generalization of the standard rejection sampling (RS) algorithm known as Ensemble Rejection Sampling (ERS). For channel simulation, we propose a new coding scheme based on ERS that achieves a near-optimal coding rate. In this process, we demonstrate that standard RS can also achieve a near-optimal coding rate and generalize the result of Braverman and Garg (2014) to the continuous alphabet setting. Next, as our main contribution, we present a distributed matching lemma for ERS, which serves as the rejection sampling counterpart to the Poisson Matching Lemma (PML) introduced by Li and Anantharam (2021). Our result also generalizes a recent work on importance matching lemma (Phan et al, 2024) and, to our knowledge, is the first result on distributed matching in the family of rejection sampling schemes where the matching probability is close to PML. We demonstrate the practical significance of our approach over prior works by applying it to distributed compression. The effectiveness of our proposed scheme is validated through experiments involving synthetic Gaussian sources and distributed image compression using the MNIST dataset.