UW Madison mathematics Colloquium is ONLINE on Fridays at 4:00 pm.

Spring 2021

January 27, 2021 [Wed 4-5pm], Morgane Austern (Microsoft Research)

(Hosted by Roch)

Asymptotics of learning on dependent and structured random objects

Classical statistical inference relies on numerous tools from probability theory to study the properties of estimators. However, these same tools are often inadequate to study modern machine problems that frequently involve structured data (e.g networks) or complicated dependence structures (e.g dependent random matrices). In this talk, we extend universal limit theorems beyond the classical setting.

Firstly, we consider distributionally “structured” and dependent random object–i.e random objects whose distribution are invariant under the action of an amenable group. We show, under mild moment and mixing conditions, a series of universal second and third order limit theorems: central-limit theorems, concentration inequalities, Wigner semi-circular law and Berry-Esseen bounds. The utility of these will be illustrated by a series of examples in machine learning, network and information theory. Secondly by building on these results, we establish the asymptotic distribution of the cross- validated risk with the number of folds allowed to grow at an arbitrary rate. Using this, we study the statistical speed-up of cross validation compared to a train-test split procedure, which reveals surprising results even when used on simple estimators.

January 29, 2021, Isaac Harris (Purdue)

(Hosted by Smith)

February 1, 2021 [Mon 4-5pm], Nan Wu (Duke)

(Hosted by Roch)

From Manifold Learning to Gaussian Process Regression on Manifolds

In this talk, I will review the concepts in manifold learning and discuss a famous manifold learning algorithm, the Diffusion Map. I will talk about my recent research results which theoretically justify that the Diffusion Map reveals the underlying topological structure of the dataset sampled from a manifold in a high dimensional space. Moreover, I will show the application of these theoretical results in solving the regression problems on manifolds and ecological problems in real life.