Generating Rectifiable Measures through Neural Networks
This provides a theoretical foundation for neural network approximation in geometric measure theory, with incremental improvements over prior work by refining the rate parameter to match rectifiability.
The paper tackles the problem of approximating rectifiable measures using neural networks, proving that m-rectifiable measures can be approximated as push-forwards of the Lebesgue measure with ReLU networks, achieving an approximation error of ε using no more than 2^b(ε) networks where b(ε) = O(ε^{-m} log^2(ε)).
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.