Georgia Institute of Technology College of Sciences

Sampling and mean-field optimization can be viewed as optimization in the space of probability distributions. Stochastic optimization algorithms like stochastic gradient descent have been immensely successful for optimization over Euclidean spaces. However, the infinite-dimensional space of probability distributions poses unique challenges. In this talk, I will discuss my recent work on the design and analysis of a stochastic algorithm for mean-field optimization with applications to the increasingly studied area of mean-field neural networks.

I'll present various methods, some old, some new,  leading to estimates on the variance of $f(X_1, X_2, \dots, X_n)$ where  

$X_1, X_2, \dots, X_n$ are independent random variables.  These methods will be illustrated with various examples.

One of the biggest discoveries in the theory of dynamical systems was that smooth (deterministic) systems can behave very randomly. Since then a rich theory of chaotic properties of smooth dynamical systems was developed using geometric, topological and probabilistic methods. In the talk we will present ideas, highlight main results and discuss techniques that were developed during the last 70 years. In the second part we plan to discuss more recent advancements and present main open questions in the field. In the last part I will focus on connections between smooth ergodic theory and number theory.

This talk explores how to construct meaningful features from noisy, high-dimensional data by leveraging geometric and invariant structures. First, we introduce a geometric framework for dimension reduction using a power-weighted path metric, which effectively de-noises high-dimensional data while preserving its intrinsic geometric structure. This framework is particularly useful for analyzing single-cell RNA data and for multi-manifold clustering, and we provide theoretical guarantees for the convergence of the associated graph Laplacian operators. We then turn to the problem of constructing features invariant to group actions in the multi-reference alignment (MRA) data model. In this setting one has many noisy observation of a hidden signal corrupted by both a group action(s) and additive noise, and one wants to recover the hidden signal from the noisy data. By formulating MRA in function space, we uncover a new connection to deconvolution: the hidden signal can be recovered from second-order Fourier statistics via an approach analogous to Kotlarski’s identity. We extend this identity to general dimensions, analyze recovery in the presence of vanishing Fourier transforms, and validate the resulting deconvolution framework with both theoretical guarantees and numerical experiments.