Monday, May 10, 2021
Time  Items 

All day 

10am 
05/10/2021  10:15am In this talk, we study the behaviour of rational points on the expanding horospheres in the space of unimodular lattices. The equidistribution of these rational points is proved by Einsiedler, Mozes, Shah and Shapira, and their proof uses techniques from homogeneous dynamics and relies in particular on measureclassification theorems due to Ratner. We pursue an alternative strategy based on Fourier analysis, spectral theory of automorphic functions and Weil’s bound for Kloosterman sums which yields an effective estimate on the rate of convergence in the space of (d+1)dimensional Euclidean lattices, for d>1. This is a joint work with D. ElBaz, B. Huang and J. Marklof. Location:
Zoom

1pm 
05/10/2021  1:00pm Abstract: Many scientific problems involve invariant structures, and learning functions that rely on a much lower dimensional set of features than the data itself. Incorporating these invariances into a parametric model can significantly reduce the model complexity, and lead to a vast reduction in the number of labeled examples required to estimate the parameters. We display this benefit in two settings. The first setting concerns ReLU networks, and the size of networks and number of points required to learn certain functions and classification regions. Here, we assume that the target function has built in invariances, namely that it only depends on the projection onto a very low dimensional, function defined manifold (with dimension possibly significantly smaller than even the intrinsic dimension of the data). We use this manifold variant of a single or multi index model to establish network complexity and ERM rates that beat even the intrinsic dimension of the data. We should note that a corollary of this result is developing intrinsic rates for a manifold plus noise data model without needing to assume the distribution of the noise decays exponentially, and we also discuss implications in twosample testing and statistical distances. The second setting for building invariances concerns linearized optimal transport (LOT), and using it to build supervised classifiers on distributions. Here, we construct invariances to families of group actions (e.g., shifts and scalings of a fixed distribution), and show that LOT can learn a classifier on group orbits using a simple linear separator. We demonstrate the benefit of this on MNIST by constructing robust classifiers with only a small number of labeled examples. This talk covers joint work with Timo Klock, Xiuyuan Cheng, and Caroline Moosmueller. email tatianna.curtis@yale.edu for info Location:
Zoom Meeting ID: 97670014308
