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 $2^{b(\varepsilon)}$ with $b(\varepsilon)=\mathcal{O}(\varepsilon^{-m}\log^2(\varepsilon))$. 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.

翻译：我们推导了（可数）$m$-可求积累度类的通用逼近结果。具体而言，我们证明利用ReLU神经网络可将$[0,1]$上的一维勒贝格测度前推逼近$m$-可求积测度，且逼近误差在Wasserstein距离下可任意小。此外，所考虑网络的权重被量化和有界，达到逼近误差$\varepsilon$所需的ReLU神经网络数量不超过$2^{b(\varepsilon)}$，其中$b(\varepsilon)=\mathcal{O}(\varepsilon^{-m}\log^2(\varepsilon))$。该结果改进了Perekrestenko等人的引理IX.4，表明当$\varepsilon$趋于零时$b(\varepsilon)$趋于无穷的速率等于可求积参数$m$，该参数可能远小于空间维度。我们将此结果推广至可数$m$-可求积累度，并证明在可数$m$-可求积支撑集的各个分量上测度满足指数衰减等技术假设时，该速率仍等于可求积参数$m$。