Joint Sparsity with Different Measurement Matrices

TL;DR

This paper extends the MMV problem to scenarios with different measurement matrices, providing probabilistic guarantees that show diverse measurement matrices can improve recovery performance even in noisy conditions.

Contribution

It introduces a generalized MMV framework with varying measurement matrices and derives recovery guarantees demonstrating benefits of measurement diversity.

Findings

Recovery probability decays exponentially with the number of measurements.

Diversity in measurement matrices can significantly enhance recovery performance.

Results are robust under bounded measurement noise.

Abstract

We consider a generalization of the multiple measurement vector (MMV) problem, where the measurement matrices are allowed to differ across measurements. This problem arises naturally when multiple measurements are taken over time, e.g., and the measurement modality (matrix) is time-varying. We derive probabilistic recovery guarantees showing that---under certain (mild) conditions on the measurement matrices---l2/l1-norm minimization and a variant of orthogonal matching pursuit fail with a probability that decays exponentially in the number of measurements. This allows us to conclude that, perhaps surprisingly, recovery performance does not suffer from the individual measurements being taken through different measurement matrices. What is more, recovery performance typically benefits (significantly) from diversity in the measurement matrices; we specify conditions under which such improvements are obtained. These results continue to hold when the measurements are subject to (bounded) noise.