Sequential Compressed Sensing

TL;DR

This paper introduces a sequential compressed sensing method that estimates reconstruction error directly from measurements, enabling adaptive stopping and performance guarantees without prior sparsity knowledge.

Contribution

It proposes a universal error estimation technique for sequential measurements, applicable to various recovery methods and signal types, improving efficiency and reliability.

Findings

Error estimates are accurate across different measurement ensembles.

Method enables adaptive measurement stopping based on confidence levels.

Applicable to both noisy and noiseless, sparse and near-sparse signals.

Abstract

Compressed sensing allows perfect recovery of sparse signals (or signals sparse in some basis) using only a small number of random measurements. Existing results in compressed sensing literature have focused on characterizing the achievable performance by bounding the number of samples required for a given level of signal sparsity. However, using these bounds to minimize the number of samples requires a-priori knowledge of the sparsity of the unknown signal, or the decay structure for near-sparse signals. Furthermore, there are some popular recovery methods for which no such bounds are known. In this paper, we investigate an alternative scenario where observations are available in sequence. For any recovery method, this means that there is now a sequence of candidate reconstructions. We propose a method to estimate the reconstruction error directly from the samples themselves, for every candidate in this sequence. This estimate is universal in the sense that it is based only on the measurement ensemble, and not on the recovery method or any assumed level of sparsity of the unknown signal. With these estimates, one can now stop observations as soon as there is reasonable certainty of either exact or sufficiently accurate reconstruction. They also provide a way to obtain "run-time" guarantees for recovery methods that otherwise lack a-priori performance bounds. We investigate both continuous (e.g. Gaussian) and discrete (e.g. Bernoulli) random measurement ensembles, both for exactly sparse and general near-sparse signals, and with both noisy and noiseless measurements.