Accuracy guarantees for L1-recovery

TL;DR

This paper introduces two new L1-based methods for sparse signal recovery from noisy data, providing verifiable performance guarantees and improved statistical properties over existing techniques like Lasso and Dantzig Selector.

Contribution

The authors propose novel recovery methods with efficiently checkable guarantees and demonstrate how to compute accuracy bounds for popular estimators, enhancing their statistical performance.

Findings

New recovery methods with performance guarantees

Efficient computation of accuracy bounds for Lasso and Dantzig Selector

Improved statistical properties over traditional estimators

Abstract

We discuss two new methods of recovery of sparse signals from noisy observation based on $\ell_1$- minimization. They are closely related to the well-known techniques such as Lasso and Dantzig Selector. However, these estimators come with efficiently verifiable guaranties of performance. By optimizing these bounds with respect to the method parameters we are able to construct the estimators which possess better statistical properties than the commonly used ones. We also show how these techniques allow to provide efficiently computable accuracy bounds for Lasso and Dantzig Selector. We link our performance estimations to the well known results of Compressive Sensing and justify our proposed approach with an oracle inequality which links the properties of the recovery algorithms and the best estimation performance when the signal support is known. We demonstrate how the estimates can be computed using the Non-Euclidean Basis Pursuit algorithm.