Purdue Probability Seminar, Fall 2023

Recovering latent communities is a key unsupervised learning task in network data with applications spanning across a multitude of disciplines. For example, identifying communities in web pages can lead to faster search, classifying regions of the human brain in communities can be used to predict onset of psychosis, and identifying communities of assets can help investors manage risk by investing in different communities of assets. However, the scale of these massive networks has become so large that it is often impossible to work with the entire network data. In this talk, I will talk about some theoretical progress for community detection in a probabilistic set up especially when we have missing data about the network. Based on joint works with Julia Gaudio, Elchanan Mossel and Colin Sandon.

The spectral heat content (SHC) measures the total heat that remains on a domain when the initial temperature is one and the outside temperature is fixed zero. When one replaces the Laplace operator in the heat equation with generators of Levy processes, one obtains SHC for those Levy processes. Recently, the two-term asymptotic behavior of SHC for isotropic stable processes on bounded C^{1,1} open sets were investigated by Park and Song (EJP 2022). In this talk, we generalize their result to cover Levy processes with regularly varying characteristic exponent with index in (1,2]. The proof provides a unified approach to the study of SHC and is applicable to both Brownian motions and jump processes. This is a joint work with Kei Kobayashi (Fordham University).

The preferential attachment model is a natural and popular random graph model for a growing network that contains very well-connected ``hubs''. We study the higher-order connectivity of such a network by investigating the topological properties of its clique complex. By determining the asymptotic growth rates of the expected Betti numbers, we discover that the graph undergoes higher-order phase transitions within the infinite-variance regime. This is joint work with Gennady Samorodnitsky, Christina Lee Yu and Rongyi He.

The purpose of dimension reduction methods for data visualization is to project high dimensional data to 2 or 3 dimensions so that humans can understand some of its structure. In this talk, we will overview some of the most popular and powerful methods in this active area. We will then the focus on two algorithms: Stochastic Neighbor Embedding (SNE) and Uniform Manifold Approximation and Projection (UMAP). Here, we will present rigorous results that establish an equilibrium distribution for these methods when the number of data points diverge in the presence of pure noise or with a planted signal. Based on joint work with Daniel Fletcher (Northwestern).

Many two-dimensional random growth models, including first-passage and last-passage percolation, are conjectured to fall within the KPZ universality class under mild assumptions on the underlying noise. In recent years, researchers have focused on a subset of exactly solvable models, where these conjectures can be rigorously verified. A wide array of methods has been employed, encompassing integrable probability, Gibbsian line ensemble, percolation arguments, and coupling techniques. This talk discusses a specific line of research that combines percolation arguments and coupling techniques to gain insights into the random geometry and space-time profiles of such growth models in the positive-temperature setting.

Random walks in random environments are Markov chains, usually with $Z^d$ as the state space, with random transition probability vectors assigned independently at each site. We study the case where there are two types of nearest-neighbor probability vectors. Most sites have the first type; we call these "blue" sites. A small fraction have the second type; we call these "red" sites. We show under appropriate assumptions that as the fraction of blue sites approaches 1, the limiting velocity of the walk converges to that of a random walk in a homogenous environment where all sites are blue. The main result is that as long as the red sites satisfy a uniform ellipticity assumption in at least two directions with fixed uniform ellipticity constant, the aforementioned convergence is uniform across all such red sites, and does not otherwise depend on what the red sites look like---they do not even need to be elliptic. The proof is based on a new coupling technique. This technique also enables us to recover a result of Kalikow from 1981, with new explicit bounds on the limiting velocity.

In this talk we discuss the Hot Spots constant for bounded smooth domains that was recently introduced by S. Steinerberger as a means to control the global extrema of the first nontrivial eigenfunction of the Neumann Laplacian by its boundary extrema. We use probabilistic techniques to derive a general formula for a dimension-dependent upper bound that can be tailored to any specific class of bounded Lipschitz domains. This formula is then used to compute upper bounds for the Hot Spots constant of the class of all bounded Lipschitz domains in $\mathbb{R}^{d}$ for both small and asymptotically large $d$ that significantly improve upon the existing results. Moreover, we prove new bounds for the Hot Spots constant for Lipschitz domains on Riemannian manifolds with non-negative Ricci curvature. This is joint work with Hugo Panzo and Jing Wang.

This talk will be about the convergence of certain symmetric $N$-particle stochastic control problems towards their mean field limits. After a brief introduction to mean field control, we will mainly discuss the following question: how fast do the value functions $V^N$ for the $N$-particle problems converge towards the value function $U$ of the mean field problem? Or in terms of partial differential equations - how fast do the solutions of certain finite-dimensional Hamilton-Jacobi equations converge to the solution of a corresponding Hamilton-Jacobi equation set on the space of probability measures? If the data is smooth and convex, then $U$ is smooth, and the rate is $O(1/N)$. When the data is not convex, $U$ may fail to be smooth, and the answer is more subtle. On one hand, it has recently been shown (in a joint work of mine with Daudin and Delarue) that if the data is smooth but not convex, the optimal global rate is $O(1/\sqrt{N})$. On the other hand, a recent paper of Cardaliaguet and Souganidis identifies an open and dense set $\mathcal O$ of initial conditions (which we call the region of strong regularity, by analogy with some classical results on first order Hamilton-Jacobi equations) where $U$ is smooth, and it is natural to wonder whether the rate of convergence might be better inside of $\mathcal O$. In an ongoing joint work with Cardaliaguet, Mimikos-Stamatopoulos, and Souganidis, we show that this is indeed the case: the rate is $O(1/N)$ locally uniformly inside the set $\mathcal O$, so the convergence is indeed faster inside $\mathcal O$ than it is outside.