Support Localization and the Fisher Metric for off-the-grid Sparse Regularization

TL;DR

This paper introduces a geometry-aware approach for off-the-grid sparse regularization using the Fisher-Rao distance, providing support recovery guarantees that are invariant under reparameterization and applicable to complex translation-varying problems.

Contribution

It demonstrates that the Fisher-Rao metric is the natural choice for support recovery in off-the-grid sparse regularization, extending guarantees to multi-dimensional and translation-varying problems.

Findings

Fisher-Rao distance ensures invariance under reparameterization.

Stable support recovery is achievable with randomized observations.

First geometry-aware guarantees for complex translation-varying problems.

Abstract

Sparse regularization is a central technique for both machine learning (to achieve supervised features selection or unsupervised mixture learning) and imaging sciences (to achieve super-resolution). Existing performance guaranties assume a separation of the spikes based on an ad-hoc (usually Euclidean) minimum distance condition, which ignores the geometry of the problem. In this article, we study the BLASSO (i.e. the off-the-grid version of $\ell^1$ LASSO regularization) and show that the Fisher-Rao distance is the natural way to ensure and quantify support recovery, since it preserves the invariance of the problem under reparameterization. We prove that under mild regularity and curvature conditions, stable support identification is achieved even in the presence of randomized sub-sampled observations (which is the case in compressed sensing or learning scenario). On deconvolution problems, which are translation invariant, this generalizes to the multi-dimensional setting existing results of the literature. For more complex translation-varying problems, such as Laplace transform inversion, this gives the first geometry-aware guarantees for sparse recovery.