A Manifold Two-Sample Test Study: Integral Probability Metric with Neural Networks
This work addresses the challenge of efficient distribution comparison in high-dimensional settings with manifold structure, which is incremental as it builds on existing IPM methods by incorporating geometric insights and neural approximations.
The paper tackles the problem of two-sample testing for high-dimensional data on low-dimensional manifolds by proposing tests based on integral probability metrics, achieving type-II risk orders such as n^{-1/max{d,2}} and n^{-(s+β)/d} depending on whether an atlas is given or not, with neural networks used to approximate Hölder functions to reduce computation.
Two-sample tests are important areas aiming to determine whether two collections of observations follow the same distribution or not. We propose two-sample tests based on integral probability metric (IPM) for high-dimensional samples supported on a low-dimensional manifold. We characterize the properties of proposed tests with respect to the number of samples $n$ and the structure of the manifold with intrinsic dimension $d$. When an atlas is given, we propose two-step test to identify the difference between general distributions, which achieves the type-II risk in the order of $n^{-1/\max\{d,2\}}$. When an atlas is not given, we propose Hölder IPM test that applies for data distributions with $(s,β)$-Hölder densities, which achieves the type-II risk in the order of $n^{-(s+β)/d}$. To mitigate the heavy computation burden of evaluating the Hölder IPM, we approximate the Hölder function class using neural networks. Based on the approximation theory of neural networks, we show that the neural network IPM test has the type-II risk in the order of $n^{-(s+β)/d}$, which is in the same order of the type-II risk as the Hölder IPM test. Our proposed tests are adaptive to low-dimensional geometric structure because their performance crucially depends on the intrinsic dimension instead of the data dimension.