# Generating Rectifiable Measures through Neural Networks

arXiv: https://arxiv.org/abs/2412.05109 · HTML: https://mlsanity.com/paper/2412.05109/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Dec 2024.

## Built on by (most cited first, 4 of 4)

- [Simultaneously Solving Infinitely Many LQ Mean Field Games In Hilbert Spaces: The Power of Neural Operators](https://mlsanity.com/paper/2510.20017/) (2025)
- [Beyond Universal Approximation Theorems: Algorithmic Uniform Approximation by Neural Networks Trained with Noisy Data](https://mlsanity.com/paper/2509.00924/) (2025)
- [Structure-Preserving Reconstruction of Convex Lipschitz Functionals on Hilbert Spaces from Finite Samples](https://mlsanity.com/paper/2605.08559/) (2026)
- [A Closed-Form Formula for Consistent Lipschitz Regression on Metric Spaces with Sparse Neural Network Realizations](https://mlsanity.com/paper/2609.03129/) (2026)

## Abstract (arXiv)

We derive universal approximation results for the class of (countably) m-rectifiable measures. Specifically, we prove that m-rectifiable measures can be approximated as push-forwards of the one-dimensional Lebesgue measure on [0,1] using ReLU neural networks with arbitrarily small approximation error in terms of Wasserstein distance. What is more, the weights in the networks under consideration are quantized and bounded and the number of ReLU neural networks required to achieve an approximation error of varepsilon is no larger than 2b(varepsilon) with b(varepsilon)=O(varepsilon-mlog2(varepsilon)). This result improves Lemma IX.4 in Perekrestenko et al. as it shows that the rate at which b(varepsilon) tends to infinity as varepsilon tends to zero equals the rectifiability parameter m, which can be much smaller than the ambient dimension. We extend this result to countably m-rectifiable measures and show that this rate still equals the rectifiability parameter m provided that, among other technical assumptions, the measure decays exponentially on the individual components of the countably m-rectifiable support set.