Analyzing Cross Validation In Compressed Sensing With Mixed Gaussian And Impulse Measurement Noise With L1 Errors

TL;DR

This paper introduces a novel use of $ ext{L}_1$ cross-validation error for compressed sensing with mixed Gaussian and impulse noise, providing theoretical guarantees and demonstrating improved robustness over traditional $ ext{L}_2$ methods.

Contribution

It offers the first theoretical analysis of $ ext{L}_1$ cross-validation in compressed sensing, showing its effectiveness in impulse noise scenarios.

Findings

$ ext{L}_1$ CV error outperforms $ ext{L}_2$ in impulse noise environments.

Choosing parameters via $ ext{L}_1$ CV minimizes recovery error with high probability.

Provides new theoretical bounds for $ ext{L}_1$ CV in CS reconstruction.

Abstract

Compressed sensing (CS) involves sampling signals at rates less than their Nyquist rates and attempting to reconstruct them after sample acquisition. Most such algorithms have parameters, for example the regularization parameter in LASSO, which need to be chosen carefully for optimal performance. These parameters can be chosen based on assumptions on the noise level or signal sparsity, but this knowledge may often be unavailable. In such cases, cross validation (CV) can be used to choose these parameters in a purely data-driven fashion. Previous work analysing the use of CV in CS has been based on the $\ell_2$ cross-validation error with Gaussian measurement noise. But it is well known that the $\ell_2$ error is not robust to impulse noise and provides a poor estimate of the recovery error, failing to choose the best parameter. Here we propose using the $\ell_1$ CV error which provides substantial performance benefits given impulse measurement noise. Most importantly, we provide a detailed theoretical analysis and error bounds for the use of $\ell_1$ CV error in CS reconstruction. We show that with high probability, choosing the parameter that yields the minimum $\ell_1$ CV error is equivalent to choosing the minimum recovery error (which is not observable in practice). To our best knowledge, this is the first paper which theoretically analyzes $\ell_1$-based CV in CS.