A Manifold Two-Sample Test Study: Integral Probability Metric with Neural Networks

TL;DR

This paper introduces new two-sample tests based on integral probability metrics that leverage neural networks to efficiently detect distribution differences in high-dimensional data supported on low-dimensional manifolds, with theoretical guarantees.

Contribution

The paper proposes neural network-based IPM tests for high-dimensional manifold-supported data, achieving optimal risk bounds and adaptivity to intrinsic data structure.

Findings

Neural network IPM test matches the risk bounds of H"older IPM.

Tests are adaptive to intrinsic low-dimensional structure.

Proposed methods are computationally feasible for high-dimensional data.

Abstract

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 H\"older IPM test that applies for data distributions with $(s,\beta)$-H\"older densities, which achieves the type-II risk in the order of $n^{-(s+\beta)/d}$. To mitigate the heavy computation burden of evaluating the H\"older IPM, we approximate the H\"older 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 $n^{-(s+\beta)/d}$, which is in the same order of the type-II risk as the H\"older 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.