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Two-sample Statistics and Distance Metrics Based on Anisotropic Kernels

Alex Cloninger


This talk introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely-many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorporating local covariance matrices and constructing an anisotropic kernel. The kernel matrix is asymmetric; it computes the affinity between n data points and a set of n_R reference points, where n_R can be drastically smaller than n.\302 While the proposed statistic can be viewed as a special class of Reproducing Kernel Hilbert Space MMD, the consistency of the test is proved, under mild assumptions of the kernel, as long as ||p-q|| ~ O(n^{-1/2+\delta}) for any \delta>0 based on a result of convergence in distribution of the test statistic. Applications to flow cytometry and diffusion MRI data sets are demonstrated, which motivate the proposed approach to compare distributions.

Tuesday, November 21, 2017
11:00AM AP&M 2402