用神经网络可精确生成规则流形测度，误差随维度降低而可控。
Generating Rectifiable Measures through Neural Networks
- 通过ReLU网络将一维勒贝格测度映射为任意m-可伸展测度
- 逼近误差ε下所需网络数不超过2的b(ε)次方，其中b(ε)=O(ε⁻ᵐ log²ε)
- 适用于低维流形结构建模，尤其适合几何结构清晰的数据
我们推导了（可数）m-可伸展测度类的通用近似结果。具体而言，证明了m-可伸展测度可通过带权值量化且有界的ReLU神经网络，作为[0,1]上一维勒贝格测度的前向推送，以任意小的瓦瑟斯坦距离误差进行逼近。当逼近误差为ε时，所需神经网络数量不超过2^{b(ε)}，其中b(ε)=O(ε^{-m} log²(ε))。该结果改进了Perekrestenko等人论文中的引理IX.4，表明b(ε)趋于无穷的速度与可伸展参数m一致，远小于环境维度。我们还将该结果扩展至可数m-可伸展测度，并在测度在各分量上指数衰减等技术假设下，证明该速率仍等于可伸展参数m。
原文摘要 · Abstract (English)
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.
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