##### Department of Mathematics,

University of California San Diego

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### Math 278C - Optimization seminar and Data Science

## Alex Cloninger

#### UCSD

## Two-sample Statistics and Distance Metrics Based on Anisotropic Kernels

##### Abstract:

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. 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 $\Vert p-q \Vert \sim$ 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 datasets are demonstrated, which motivate the proposed approach to compare distributions.

Jiawang Nie

### October 11, 2017

### 4:00 PM

### AP&M 2402

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