Stable recovery of low-dimensional cones in Hilbert spaces: One RIP to rule them all

TL;DR

This paper develops a unified RIP-based framework for stable recovery of low-dimensional cones in Hilbert spaces, extending guarantees to arbitrary regularizers and demonstrating improved results for structured sparsity.

Contribution

It introduces generic RIP-based guarantees for cone recovery with any regularizer, unifying and strengthening existing results in inverse problems.

Findings

Established RIP guarantees for arbitrary regularizers.

Improved recovery guarantees for block structured sparsity.

Validated theoretical results with selected examples.

Abstract

Many inverse problems in signal processing deal with the robust estimation of unknown data from underdetermined linear observations. Low dimensional models, when combined with appropriate regularizers, have been shown to be efficient at performing this task. Sparse models with the 1-norm or low rank models with the nuclear norm are examples of such successful combinations. Stable recovery guarantees in these settings have been established using a common tool adapted to each case: the notion of restricted isometry property (RIP). In this paper, we establish generic RIP-based guarantees for the stable recovery of cones (positively homogeneous model sets) with arbitrary regularizers. These guarantees are illustrated on selected examples. For block structured sparsity in the infinite dimensional setting, we use the guarantees for a family of regularizers which efficiency in terms of RIP constant can be controlled, leading to stronger and sharper guarantees than the state of the art.