# Infinite dimensional generative sensing

arXiv: https://arxiv.org/abs/2603.03196 · HTML: https://mlsanity.com/paper/2603.03196/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 46 references that are in the map.
- First posted on arXiv 3 Mar 2026.

## Builds on (most cited first, 19 of 19)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Deep Unsupervised Learning using Nonequilibrium Thermodynamics](https://mlsanity.com/paper/1503.03585/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [A Survey on Diffusion Models for Inverse Problems](https://mlsanity.com/paper/2410.00083/) (2024)
- [A Mathematical Guide to Operator Learning](https://mlsanity.com/paper/2312.14688/) (2023)
- [∞-Diff: Infinite Resolution Diffusion with Subsampled Mollified States](https://mlsanity.com/paper/2303.18242/) (2023)
- [On efficient algorithms for computing near-best polynomial approximations to high-dimensional, Hilbert-valued functions from limited samples](https://mlsanity.com/paper/2203.13908/) (2022)
- [A coherence parameter characterizing generative compressed sensing with Fourier measurements](https://mlsanity.com/paper/2207.09340/) (2022)
- [Autoencoders in Function Space](https://mlsanity.com/paper/2408.01362/) (2024)
- [Continuous Generative Neural Networks: A Wavelet-Based Architecture in Function Spaces](https://mlsanity.com/paper/2205.14627/) (2022)
- [Model-adapted Fourier sampling for generative compressed sensing](https://mlsanity.com/paper/2310.04984/) (2023)
- [Optimal sampling for least-squares approximation](https://mlsanity.com/paper/2409.02342/) (2024)
- [A unified framework for learning with nonlinear model classes from arbitrary linear samples](https://mlsanity.com/paper/2311.14886/) (2023)
- [Compressed sensing for inverse problems and the sample complexity of the sparse Radon transform](https://mlsanity.com/paper/2302.03577/) (2023)
- [Compressed sensing for inverse problems II: applications to deconvolution, source recovery, and MRI](https://mlsanity.com/paper/2501.01929/) (2025)

## Abstract (arXiv)

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.