# Recovery guarantees for compressed sensing with unknown errors

## TL;DR

This paper establishes robust recovery guarantees for l1-minimization in compressed sensing without requiring prior noise estimates, enhancing practical applicability in high-dimensional and inverse problem contexts.

## Contribution

It provides novel theoretical guarantees for l1-minimization robustness when noise estimates are unknown, applicable to various high-dimensional and inverse problem scenarios.

## Key findings

- Robust recovery guarantees for basis pursuit without noise estimates.
- Application to high-dimensional function approximation and inverse problems.
- Relevance to MRI and other imaging techniques.

## Abstract

From a numerical analysis perspective, assessing the robustness of l1-minimization is a fundamental issue in compressed sensing and sparse regularization. Yet, the recovery guarantees available in the literature usually depend on a priori estimates of the noise, which can be very hard to obtain in practice, especially when the noise term also includes unknown discrepancies between the finite model and data. In this work, we study the performance of l1-minimization when these estimates are not available, providing robust recovery guarantees for quadratically constrained basis pursuit and random sampling in bounded orthonormal systems. Several applications of this work are approximation of high-dimensional functions, infinite-dimensional sparse regularization for inverse problems, and fast algorithms for non-Cartesian Magnetic Resonance Imaging.