Bharath Sriperumbudur is an associate professor in the Department of Statistics at the Pennsylvania State University. His research interests include non-parametric statistics, machine learning, statistical learning theory, optimal transport and gradient flows, regularization and inverse problems, reproducing kernel spaces in probability and statistics, functional and topological data analysis.
His current research is supported by the award NSF-DMS-CAREER-1945396, "Statistical Learning, Inference and Approximation with Reproducing Kernels."
Talk: Spectral Regularized Kernel Hypothesis Tests
Abstract: Over the last decade, an approach that has gained a lot of popularity in tackling non-parametric testing problems on general (i.e., non-Euclidean) domains is based on the notion of reproducing kernel Hilbert space (RKHS) embedding of probability distributions. The main goal of our work is to understand the optimality of two-sample and goodness-of-fit tests constructed based on this approach. First, we show that the popular MMD (maximum mean discrepancy) based hypothesis tests are not optimal in terms of the separation boundary measured in Hellinger distance. Second, we propose a modification to the MMD test based on spectral regularization by taking into account the covariance information (which is not captured by the MMD test) and prove the proposed test to be minimax optimal with a smaller separation boundary than that achieved by the MMD test. Third, we propose an adaptive version of the above test, which involves a data-driven strategy to choose the regularization parameter and show the adaptive test to be almost minimax optimal up to a logarithmic factor. Moreover, our results hold for the permutation variant of the test where the test threshold is chosen elegantly through the permutation of the samples. Through numerical experiments on synthetic and real-world data, we demonstrate the superior performance of the proposed test in comparison to many popular tests.
(Based on joint work with Omar Hagrass (PSU) and Bing Li (PSU))