Title: Constructing Rectifiable Measures via Neural Networks

Abstract: This study establishes universal approximation theorems for the family of (countably) $m$-rectifiable measures. We demonstrate that such measures can be approximated by mapping the one-dimensional Lebesgue measure defined on the interval $[0,1]$ through ReLU neural networks, achieving arbitrarily low error rates as measured by the Wasserstein distance. Furthermore, the neural networks employed feature weights that are both quantized and bounded. To attain an approximation error of $\varepsilon$, the quantity of ReLU neural networks needed does not exceed $2^{b(\varepsilon)}$, where $b(\varepsilon)=\mathcal{O}(\varepsilon^{-m}\log^2(\varepsilon))$. This finding enhances Lemma IX.4 from Perekrestenko et al. by revealing that the growth rate of $b(\varepsilon)$ as $\varepsilon$ approaches zero is determined by the rectifiability parameter $m$. This parameter can be significantly smaller than the ambient dimension. Additionally, we generalize these findings to countably $m$-rectifiable measures, confirming that the aforementioned rate remains equal to $m$, contingent upon several technical conditions, including the requirement that the measure exhibits exponential decay across the individual components of the countably $m$-rectifiable support set.

Source: arXiv Generated at: 2026-06-03 00:00:00 UTC