# Denoising guarantees for optimized sampling schemes in compressed sensing

arXiv: https://arxiv.org/abs/2504.01046 · HTML: https://mlsanity.com/paper/2504.01046/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 29 references that are in the map.
- First posted on arXiv 1 Apr 2025.

## Builds on (most cited first, 8 of 8)

- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [Real or Not Real, that is the Question](https://mlsanity.com/paper/2002.05512/) (2020)
- [CS4ML: A general framework for active learning with arbitrary data based on Christoffel functions](https://mlsanity.com/paper/2306.00945/) (2023)
- [A coherence parameter characterizing generative compressed sensing with Fourier measurements](https://mlsanity.com/paper/2207.09340/) (2022)
- [CAS4DL: Christoffel Adaptive Sampling for function approximation via Deep Learning](https://mlsanity.com/paper/2208.12190/) (2022)
- [Model-adapted Fourier sampling for generative compressed sensing](https://mlsanity.com/paper/2310.04984/) (2023)
- [Beyond Independent Measurements: General Compressed Sensing with GNN Application](https://mlsanity.com/paper/2111.00327/) (2021)
- [A unified framework for learning with nonlinear model classes from arbitrary linear samples](https://mlsanity.com/paper/2311.14886/) (2023)

## Built on by (most cited first, 2 of 2)

- [Partially deterministic sampling for compressed sensing with denoising guarantees](https://mlsanity.com/paper/2604.04802/) (2026)
- [Active Learning for Conditional Generative Compressed Sensing](https://mlsanity.com/paper/2605.05435/) (2026)

## Abstract (arXiv)

Compressed sensing with subsampled unitary matrices benefits from optimized sampling schemes, which feature improved theoretical guarantees and empirical performance relative to uniform subsampling. We provide, in a first of its kind in compressed sensing, theoretical guarantees showing that the error caused by the measurement noise vanishes with an increasing number of measurements for optimized sampling schemes, assuming that the noise is Gaussian. We moreover provide similar guarantees for measurements sampled with-replacement with arbitrary probability weights. All our results hold on prior sets contained in a union of low-dimensional subspaces. Finally, we demonstrate that this denoising behavior appears in empirical experiments with a rate that closely matches our theoretical guarantees when the prior set is the range of a generative ReLU neural network and when it is the set of sparse vectors.