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The prolonged COVID-19 pandemic has tied up significant medical resources, and its management poses a challenge for the public health care decision making. Accurate predictions of the hospitalizations are crucial for the decision makers to make informed decision for the medical resource allocation. This paper proposes a method named County Augmented Transformer (CAT). To generate accurate predictions of four-week-ahead COVID-19 related hospitalizations for every states in the United States. Inspired by the modern deep learning techniques, our method is based on a self-attention model (known as the transformer model) that is actively used in Natural Language Processing. Our transformer based model can capture both short-term and long-term dependencies within the time series while enjoying computational efficiency. Our model is a data based approach that utilizes the publicly available information including the COVID-19 related number of confirmed cases, deaths, hospitalizations data, and the household median income data. Our numerical experiments demonstrate the strength and the usability of our model as a potential tool for assisting the medical resources allocation.

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.