Uniform Recovery from Subgaussian Multi-Sensor Measurements

TL;DR

This paper establishes uniform recovery guarantees for compressed sensing in multi-sensor systems with subgaussian sampling, showing that the number of measurements per sensor can decrease linearly with the number of sensors, improving stability and robustness.

Contribution

The paper introduces new uniform recovery guarantees for multi-sensor compressed sensing systems using subgaussian measurements, with explicit conditions and simplified constants.

Findings

Optimal measurement scaling decreases linearly with the number of sensors.

Established the Asymmetric Restricted Isometry Property (ARIP) for multi-sensor systems.

Unified block-diagonal sensing matrices as a special case with comparable guarantees.

Abstract

Parallel acquisition systems are employed successfully in a variety of different sensing applications when a single sensor cannot provide enough measurements for a high-quality reconstruction. In this paper, we consider compressed sensing (CS) for parallel acquisition systems when the individual sensors use subgaussian random sampling. Our main results are a series of uniform recovery guarantees which relate the number of measurements required to the basis in which the solution is sparse and certain characteristics of the multi-sensor system, known as sensor profile matrices. In particular, we derive sufficient conditions for optimal recovery, in the sense that the number of measurements required per sensor decreases linearly with the total number of sensors, and demonstrate explicit examples of multi-sensor systems for which this holds. We establish these results by proving the so-called Asymmetric Restricted Isometry Property (ARIP) for the sensing system and use this to derive both nonuniversal and universal recovery guarantees. Compared to existing work, our results not only lead to better stability and robustness estimates but also provide simpler and sharper constants in the measurement conditions. Finally, we show how the problem of CS with block-diagonal sensing matrices can be viewed as a particular case of our multi-sensor framework. Specializing our results to this setting leads to a recovery guarantee that is at least as good as existing results.