# Compressed sensing in Hilbert spaces

## TL;DR

This paper develops a general framework for compressed sensing in high- or infinite-dimensional Hilbert spaces, showing how random measurements enable the recovery of low-dimensional structured signals through regularization techniques.

## Contribution

It introduces a unified approach to analyze recovery guarantees in Hilbert spaces using random measurements and regularization methods.

## Key findings

- Random measurements guarantee signal recovery under certain conditions
- Framework applies to high-dimensional and infinite-dimensional spaces
- Performance of regularized recovery methods can be systematically studied

## Abstract

In many linear inverse problems, we want to estimate an unknown vector belonging to a high-dimensional (or infinite-dimensional) space from few linear measurements. To overcome the ill-posed nature of such problems, we use a low-dimension assumption on the unknown vector: it belongs to a low-dimensional model set. The question of whether it is possible to recover such an unknown vector from few measurements then arises. If the answer is yes, it is also important to be able to describe a way to perform such a recovery. We describe a general framework where appropriately chosen random measurements guarantee that recovery is possible. We further describe a way to study the performance of recovery methods that consist in the minimization of a regularization function under a data-fit constraint.