# Model-adapted Fourier sampling for generative compressed sensing

arXiv: https://arxiv.org/abs/2310.04984 · HTML: https://mlsanity.com/paper/2310.04984/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 23 references that are in the map.
- First posted on arXiv 8 Oct 2023.

## Builds on (most cited first, 9 of 9)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Compressed Sensing using Generative Models](https://mlsanity.com/paper/1703.03208/) (2017)
- [A Statistical Perspective on Algorithmic Leveraging](https://mlsanity.com/paper/1306.5362/) (2013)
- [Fourier Sparse Leverage Scores and Approximate Kernel Learning](https://mlsanity.com/paper/2006.07340/) (2020)
- [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)

## Built on by (most cited first, 6 of 6)

- [A unified framework for learning with nonlinear model classes from arbitrary linear samples](https://mlsanity.com/paper/2311.14886/) (2023)
- [Denoising guarantees for optimized sampling schemes in compressed sensing](https://mlsanity.com/paper/2504.01046/) (2025)
- [How many measurements are enough? Bayesian recovery in inverse problems with general distributions](https://mlsanity.com/paper/2505.10630/) (2025)
- [Infinite dimensional generative sensing](https://mlsanity.com/paper/2603.03196/) (2026)
- [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)

We study generative compressed sensing when the measurement matrix is randomly subsampled from a unitary matrix (with the DFT as an important special case). It was recently shown that textitO(kdn| boldsymbolα|∞2) uniformly random Fourier measurements are sufficient to recover signals in the range of a neural network G:ℝk → ℝn of depth d, where each component of the so-called local coherence vector boldsymbolα quantifies the alignment of a corresponding Fourier vector with the range of G. We construct a model-adapted sampling strategy with an improved sample complexity of textitO(kd| boldsymbolα|22) measurements. This is enabled by: (1) new theoretical recovery guarantees that we develop for nonuniformly random sampling distributions and then (2) optimizing the sampling distribution to minimize the number of measurements needed for these guarantees. This development offers a sample complexity applicable to natural signal classes, which are often almost maximally coherent with low Fourier frequencies. Finally, we consider a surrogate sampling scheme, and validate its performance in recovery experiments using the CelebA dataset.