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Abstract Most COVID-19 studies commonly report figures of the overall infection at a state- or county-level. This aggregation tends to miss out on fine details of virus propagation. In this paper, we analyze a high-resolution COVID-19 dataset in Cali, Colombia, that records the precise time and location of every confirmed case. We develop a non-stationary spatio-temporal point process equipped with a neural network-based kernel to capture the heterogeneous correlations among COVID-19 cases. The kernel is carefully crafted to enhance expressiveness while maintaining model interpretability. We also incorporate some exogenous influences imposed by city landmarks. Our approach outperforms the state-of-the-art in forecasting new COVID-19 cases with the capability to offer vital insights into the spatio-temporal interaction between individuals concerning the disease spread in a metropolis. 

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 a 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 $$(s,\beta )$$-Hölder densities, which achieves the type-II risk in the order of $$n^{-(s+\beta )/d}$$. 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 $$n^{-(s+\beta )/d}$$, 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.