Wednesday, November 17, 2021
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All day |
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2pm |
11/17/2021 - 2:30pm Abstract: Kernel-based test is widely used non-parametric statistic to compare two distributions, particularly multivariate data distributions. However, such kernel-based test, notably the Maximum Mean Discrepancy (MMD), is known to face difficulties for high-dimensional data as well as computational challenges in practice. This talk will start from analyzing kernel tests applied to low-dimensional manifold data embedded in high dimensional space, and theoretically show that curse-of-dimensionality can be automatically avoided using local kernels. Our non-asymptotic result proves test power at finite sample size, and holds for a class of regular and decay kernel functions that are not necessarily positive semi-definite. We then discuss the practical challenges of kernel tests, primarily the choice of kernel bandwidth and the computational bottleneck. For the former, we show a recent analysis of k-nearest-neighbor self-tuned kernel which provably reduces variance error and improves the stability of kernel methods at places where data density can be low (joint work with Hau-Tieng Wu, Duke). For the latter, we revisit neural network classification two-sample tests, which show empirical advantage yet lack full theoretical understanding, especially that of a trained neural network. To the end of understanding training dynamics of neural network two-sample tests, we introduce neural tangent kernel (NTK) MMD, which provably approximates kernel MMD of a finite-width NTK and consequently enjoys theoretical kernel test power guarantee. In practice, NTK-MMD can be computed from small-batch one-pass stochastic gradient descent on the training split, and allows calibration of test threshold via test-split-only bootstrap (thus avoiding evaluating network gradients on the test samples). Joint work with Yao Xie, Georgia Tech. Location:
https://yale.zoom.us/j/2188028533
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4pm |
11/17/2021 - 4:00pm The Putnam seminar meets every Wednesday from 4 to 5:30 in LOM 214. As always, everyone is warmly welcomed to come to hang out, learn more cool math, and meet folks. The seminar is casual, and folks can come and go as they like. See Pat Devlin’s webpage (and/or contact him) for more information. Folks can sign up for the mailing list here: https://forms.gle/nYPx72KVJxJcgLha8 Location:
LOM 214
11/17/2021 - 4:15pm Abstract: It was already known to Klein that the modular group SL_2(Z) has Location: |