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Infinite dimensional generative sensing

Paolo Angella, Vito Paolo Pastore, Matteo Santacesaria

TL;DR

This work presents a rigorous framework for generative compressed sensing in Hilbert spaces, and extends the notion of local coherence in an infinite-dimensional setting, to derive optimal, resolution-independent sampling distributions.

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.

Infinite dimensional generative sensing

TL;DR

This work presents a rigorous framework for generative compressed sensing in Hilbert spaces, and extends the notion of local coherence in an infinite-dimensional setting, to derive optimal, resolution-independent sampling distributions.

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.
Paper Structure (23 sections, 11 theorems, 72 equations, 6 figures)

This paper contains 23 sections, 11 theorems, 72 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Structure of the paper
  4. Setup and Main Results
  5. Local Coherence and Sampling Strategies
  6. Recovery Guarantees
  7. The Balancing Property
  8. Numerical Validation
  9. Experimental Setup and Methodology
  10. Dataset and Architecture
  11. Training
  12. Fitting the Latent Space
  13. Reconstruction
  14. Results
  15. Local coherence and adaptive sampling
  16. ...and 8 more sections

Key Result

Lemma 2.1

Let $F \colon \ell^2(\mathbb{N}) \longmapsto \ell^2(\mathbb{N})$ be a unitary operator and let $\mathcal{U}\subseteq\ell^2(\mathbb{N})$ be a finite union of convex cones contained in a subspace of dimension $r$, $\alpha=(\alpha_i)_{i\in\mathbb{N}}$ the local coherence of $\mathcal{U}$ as defined in

Figures (6)

  • Figure 1: Comparison between the original maximum-magnitude based coherence estimator and the self-difference based estimator. Although the latter is theoretically better aligned with the definition of local coherence, the former yields superior empirical performance (model trained and with output in 64 x 64 resolution).
  • Figure 2: Reconstruction error versus sampling rate for uniform and adaptive Fourier sampling.
  • Figure 3: Reconstruction examples comparing uniform and adaptive sampling at different rates (model trained on 64 x 64 resolution).
  • Figure 4: Recovery error across different training and output discretizations.
  • Figure 5: Example reconstructions for varying output resolutions (model trained on 64 x 64 resolution)
  • ...and 1 more figures

Theorems & Definitions (32)

  • Definition 2.1: $(k,d)$-Generalized Generative Network
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.1
  • Definition 2.6
  • Lemma 2.2
  • Theorem 2.3
  • Remark
  • ...and 22 more