Information-Geometric Analysis of the Optimal Error Exponent in Fixed-Length Hypothesis Testing

Hypothesis testing has emerged as a significant research area due to its applications in various domains. In many real-world scenarios, we can only obtain training samples of both hypotheses instead of the underlying distributions, and the decision rule shall be restricted to the form of testing sample statistics due to the computational requirement of highdimensional data. In this paper, we study the fixed-length hypothesis testing problem under such constraints. By applying the information-geometric method, we provide the corresponding asymptotic optimal error exponent with respect to the sequence lengths, and propose a valid decision rule. Moreover, we present a geometric interpretation of the trade-off between the sampling processes of training and testing.