• Speaker: Zhenqi Wang, Michigan State University
• Title: Smooth local rigidity for hyperbolic toral automorphisms
• Date: 06/07/2022
• Time: 2:00 PM - 3:00 PM
• Place: C304 Wells Hall
• Contact: Huyi Hu (hhu@msu.edu)

We study the regularity of a conjugacy $H$ between a hyperbolic toral automorphism $A$ and its smooth perturbation $f$. We show that if $H$ is weakly differentiable then it is $C^{1+\text{Holder}}$ and, if $A$ is also weakly irreducible, then $H$ is $C^\infty$. As a part of the proof, we establish results of independent interest on Holder continuity of a measurable conjugacy between linear cocycles over a hyperbolic system. As a corollary, we improve regularity of the conjugacy to $C^\infty$ in prior local rigidity results. This is a joint work with B. Kalinin an V. Sadovskaya.

• Speaker: Fanny Yang, ETH Zurich
• Title: Fast rates for noisy interpolation require rethinking the effects of inductive bias
• Date: 06/09/2022
• Time: 4:30 AM - 4:30 AM
• Place: Online (virtual meeting) (Virtual Meeting Link)
• Contact: Olga Turanova (turanova@msu.edu)

Modern machine learning has uncovered an interesting observation: large over parameterized models can achieve good generalization performance despite interpolating noisy training data. In this talk, we study high-dimensional linear models and show how interpolators can achieve fast statistical rates when their structural bias is moderate. More concretely, while minimum-l2-norm interpolators cannot recover the signal in high dimensions, minimum-l1-interpolators with strong sparsity bias are much more sensitive to noise. In fact, we show that even though they are asymptotically consistent, minimum-l1-norm interpolators converge with a logarithmic rate much slower than the O(1/n) rate of regularized estimators. In contrast, minimum-lp-norm interpolators with 1<p<2 can trade off these two competing trends to yield polynomial rates close to O(1/n).