The Bethe–Hessian matrix, introduced by Saade, Krzakala, and Zdeborová (2014), is a Hermitian operator tailored for spectral clustering on sparse networks. Unlike the non-symmetric, high-dimensional non-backtracking operator, this matrix is conjectured to reach the Kesten–Stigum threshold in the sparse stochastic block model (SBM), yet a fully rigorous analysis of the method has remained open. Beyond its practical utility, this sparse random matrix exhibits a surprising phenomenon called "one-sided eigenvector localization" that has not been fully explained.

We present the first rigorous analysis of the Bethe–Hessian spectral method for the SBM and partially answer some open questions in Saade, Krzakala, and Zdeborová (2014).  Joint work with Ludovic Stephan.

Is there a natural way to order data in dimension greater than one? The approach based on the notion of data depth, often associated with the name of John Tukey, is among the most popular. Tukey’s depth has found applications in robust statistics, the study of elections and social choice, and graph theory. We will give an introduction to the topic (with an emphasis on robust statistics), describe some remaining open questions as well as our recent progress towards the solutions.

This talk is based on the joint work with Yinan Shen.

We will discuss recent advances in the compression of pre-trained neural networks using both novel and existing computationally efficient algorithms. The approaches we consider leverage sparsity, low-rank approximations of weight matrices, and weight quantization to achieve significant reductions in model size, while maintaining performance. We provide rigorous theoretical error guarantees as well as numerical experiments.

In this talk, we present a new proof of the delocalization conjecture for random band matrices. The conjecture asserts that for an $N\times N$ random matrix with bandwidth $W$, the eigenvectors in the bulk of the spectrum are delocalized provided that $W \gg N^{1/2}$. Moreover, in this regime, the eigenvalue distribution aligns with that of Gaussian random ensembles (i.e., GOE or GUE). Our proof employs a novel loop hierarchy method and leverages the sum-zero property, a concept that was instrumental in the previous work on high-dimensional random matrices. 

This work is a joint collaboration with H.T. Yau.

I will discuss adapted transport theory and the adapted Wasserstein distance, which extend classical transport theory from probability measures to stochastic processes by incorporating the temporal flow of information. This adaptation addresses key limitations of classical transport when dealing with time-dependent data. 

I will highlight how, unlike other topologies for stochastic processes, the adapted Wasserstein distance ensures continuity for fundamental probabilistic operations, including the Doob decomposition, optimal stopping, and stochastic control. Additionally, I will explore how adapted transport preserves many desirable properties of classical transport theory, making it a powerful tool for analyzing stochastic systems.