The basins of attraction of the global minimizers of non-convex inverse problems with low-dimensional models in infinite dimension

TL;DR

This paper develops a theoretical framework for understanding the basins of attraction in non-convex inverse problems with low-dimensional models, linking basin size to measurement count and extending known results to new Gaussian mixture estimation scenarios.

Contribution

It introduces a unified framework for both finite and infinite dimensional inverse problems, providing new insights into basin sizes and measurement requirements.

Findings

Basin size is related to the number of measurements.

Framework recovers known results for low-rank and sparse estimation.

Provides new results for Gaussian mixture estimation.

Abstract

Non-convex methods for linear inverse problems with low-dimensional models have emerged as an alternative to convex techniques. We propose a theoretical framework where both finite dimensional and infinite dimensional linear inverse problems can be studied. We show how the size of the the basins of attraction of the minimizers of such problems is linked with the number of available measurements. This framework recovers known results about low-rank matrix estimation and off-the-grid sparse spike estimation, and it provides new results for Gaussian mixture estimation from linear measurements. keywords: low-dimensional models, non-convex methods, low-rank matrix recovery, off-the-grid sparse recovery, Gaussian mixture model estimation from linear measurements.