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.

Abstract:

It was already known to Klein that the modular group SL_2(Z) has
extraordinarily many finite index subgroups, most of which are not defined
by congruence conditions on the matrix entries. The Unbounded Denominators
conjecture, solved in a recent joint work with Frank Calegari and Yunqing
Tang, is an amendment for the failure of the Congruence Subgroup Property
for SL_2(Z): a Z[[q]] condition on the Fourier expansions of the modular
forms living on a given finite index subgroup of SL_2(Z) turns out to be a
necessary and sufficient condition for the subgroup to be congruence.

Algebra and Geometry lecture series (yale.edu)

Abstract:

We describe some recent work that has been done to generalize the notion of lower scalar curvature bounds to C^0 metrics, including a localized Ricci flow approach. In particular, we show the following: that our definition is stable under greater-than-second-order perturbation of the metric, that there exists a reasonable notion of a Ricci flow starting from C^0 initial data which is smooth for positive times, and that the weak lower scalar curvature bounds are preserved under evolution by the Ricci flow from C^0 initial data.