Subsampling at Information Theoretically Optimal Rates

TL;DR

This paper introduces a novel sampling scheme inspired by coding theory that acquires a small random subset of Gabor coefficients, achieving near-optimal reconstruction rates at the information theoretic limit for sparse frequency signals.

Contribution

It proposes a new probabilistic sampling method using Gabor coefficients that reaches the theoretical minimum sampling rate for accurate reconstruction.

Findings

Achieves correct reconstruction at the information theoretic limit.

Empirical evidence supports the scheme's optimality.

Uses a small random subset of Gabor coefficients for sampling.

Abstract

We study the problem of sampling a random signal with sparse support in frequency domain. Shannon famously considered a scheme that instantaneously samples the signal at equispaced times. He proved that the signal can be reconstructed as long as the sampling rate exceeds twice the bandwidth (Nyquist rate). Cand\`es, Romberg, Tao introduced a scheme that acquires instantaneous samples of the signal at random times. They proved that the signal can be uniquely and efficiently reconstructed, provided the sampling rate exceeds the frequency support of the signal, times logarithmic factors. In this paper we consider a probabilistic model for the signal, and a sampling scheme inspired by the idea of spatial coupling in coding theory. Namely, we propose to acquire non-instantaneous samples at random times. Mathematically, this is implemented by acquiring a small random subset of Gabor coefficients. We show empirically that this scheme achieves correct reconstruction as soon as the sampling rate exceeds the frequency support of the signal, thus reaching the information theoretic limit.