2022/05/04 by Jie Wang, Minshuo Chen, Wang, Jie +7 · 1 citation
Mathematics · #Advanced Statistical Methods and Models #FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Statistical Methods and Bayesian Inference #Statistical Methods and Inference #Statistics Theory (math.ST)
paper · pdf · doi:10.48550/arxiv.2205.02043
openalex publication_date 2022/05/04 · openalex created_date 2022/05/08 · openalex updated_date 2026/07/28
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.