Infinite dimensional generative sensing

TL;DR

This paper develops a theoretical framework for infinite-dimensional generative compressed sensing in Hilbert spaces, establishing stable recovery guarantees and demonstrating practical benefits through numerical experiments on the Darcy flow equation.

Contribution

It extends generative compressed sensing theory to infinite-dimensional Hilbert spaces, introducing new notions of coherence and RIP, and validates the approach with numerical experiments.

Findings

Stable recovery when measurements are proportional to intrinsic dimension

Resolution-independent sampling distributions derived

Lower-resolution generators improve stability in undersampled regimes

Abstract

Deep generative models have become a standard for modeling priors for inverse problems, going beyond classical sparsity-based methods. However, existing theoretical guarantees are mostly confined to finite-dimensional vector spaces, creating a gap when the physical signals are modeled as functions in Hilbert spaces. This work presents a rigorous framework for generative compressed sensing in Hilbert spaces. We extend the notion of local coherence in an infinite-dimensional setting, to derive optimal, resolution-independent sampling distributions. Thanks to a generalization of the Restricted Isometry Property, we show that stable recovery holds when the number of measurements is proportional to the prior's intrinsic dimension (up to logarithmic factors), independent of the ambient dimension. Finally, numerical experiments on the Darcy flow equation validate our theoretical findings and demonstrate that in severely undersampled regimes, employing lower-resolution generators acts as an implicit regularizer, improving reconstruction stability.