Jonathan Leake

Jonathan Leake, Shayan Oveis Gharan

Let $μ$ be a probability distribution on a multi-state spin system on a set $V$ of sites; equivalently, a $d$-partite simplicial complex with distribution $μ$ on maximal faces. For any pair of vertices $u,v\in V$, define the pairwise spectral influence $\mathcal{I}_{u,v}$ as follows. Let $σ$ be a choice of spins $s_w\in S_w$ for every $w\in V\setminus\{u,v\}$, and construct a matrix in $\mathbb{R}^{(S_u\cup S_v)\times (S_u\cup S_v)}$ where for any $s_u\in S_u, s_v\in S_v$, the $(us_u,vs_v)$-entry is the probability that $s_v$ is the spin of $v$ conditioned on $s_u$ being the spin of $u$ and on $σ$. Then $\mathcal{I}_{u,v}$ is the maximal second eigenvalue of this matrix, over all choices of spins for all $w\in V\setminus\{u,v\}$. Equivalently, $\mathcal{I}_{u,v}$ is the maximum local spectral expansion of links of codimension $2$ that include a spin for every $w \in V \setminus \{u,v\}$. We show that if the largest eigenvalue of the pairwise spectral influence matrix with entries $\mathcal{I}_{u,v}$ is bounded away from 1, i.e. $λ_{\max}(\mathcal{I})\leq 1-ε$ (and $X$ is connected), then the Glauber dynamics mixes rapidly and generate samples from $μ$. This improves/generalizes the classical Dobrushin's influence matrix as the $\mathcal{I}_{u,v}$ lower-bounds the classical influence of $u\to v$. As an application, we prove that the Glauber dynamics mixes rapidly up to (approximately) the phase transition for the multi-state hardcore model--a widely studied model in telecommunication networks and statistical physics (generalizing the hardcore model) introduced by Mazel and Suhov. As a by-product of our results, we also prove improved/almost optimal trickle-down theorems for partite simplicial complexes. Our proof builds on the trickle-down theorems via $\mathcal{C}$-Lorentzian polynomials machinery recently developed by the authors and Lindberg.

Jonathan Leake, Nisheeth K. Vishnoi

In the last few years, the notion of symmetry has provided a powerful and essential lens to view several optimization or sampling problems that arise in areas such as theoretical computer science, statistics, machine learning, quantum inference, and privacy. Here, we present two examples of nonconvex problems in optimization and sampling where continuous symmetries play -- implicitly or explicitly -- a key role in the development of efficient algorithms. These examples rely on deep and hidden connections between nonconvex symmetric manifolds and convex polytopes, and are heavily generalizable. To formulate and understand these generalizations, we then present an introduction to Lie theory -- an indispensable mathematical toolkit for capturing and working with continuous symmetries. We first present the basics of Lie groups, Lie algebras, and the adjoint actions associated with them, and we also mention the classification theorem for Lie algebras. Subsequently, we present Kostant's convexity theorem and show how it allows us to reduce linear optimization problems over orbits of Lie groups to linear optimization problems over polytopes. Finally, we present the Harish-Chandra and the Harish-Chandra--Itzykson--Zuber (HCIZ) formulas, which convert partition functions (integrals) over Lie groups into sums over the corresponding (discrete) Weyl groups, enabling efficient sampling algorithms.