Sequential Sweeps and High Dimensional Expansion
Vedat Levi Alev, Ori Parzanchevski
TL;DR
The paper analyzes the sequential sweep Glauber dynamics on $n$-partite simplicial complexes, comparing it to the traditional down-up walk and showing that, under strong local expansion, the sequential sweep achieves rapid mixing even when the down-up walk is bottlenecked. By developing a geometric, projection-based framework and introducing the $\vec\varepsilon$-product and entropy-contraction notions, the authors derive explicit bounds on the spectral gap and entropy contraction of $P_{\mathtt{seq}}$, linking them to the local structure of the complex via colored random walks. They establish a sharp bound on $\sigma_2(\mathsf P_{\mathtt{seq}})$ in terms of $\varepsilon^{I \to J}$ and show that, under completely $\varepsilon$-product or suitably connected-link assumptions, the gap approaches 1 as the local expansion improves. An entropy-contraction inequality for $P_{\mathtt{seq}}$ yields strong KL-divergence contraction and accompanying mixing-time bounds, with broader implications for sampling and concentration in high-dimensional combinatorial structures. The work integrates high-dimensional expansion techniques, spectral-independence concepts, and entropic methods to advance understanding of systematic-scan dynamics in complex product spaces.
Abstract
It is well known that the spectral gap of the down-up walk over an $n$-partite simplicial complex (also known as Glauber dynamics) cannot be better than $O(1/n)$ due to natural obstructions such as coboundaries. We study an alternative random walk over partite simplicial complexes known as the sequential sweep or the systematic scan Glauber dynamics: Whereas the down-up walk at each step selects a random coordinate and updates it based on the remaining coordinates, the sequential sweep goes through each of the coordinates one by one in a deterministic order and applies the same update operation. It is natural, thus, to compare $n$-steps of the down-up walk with a single step of the sequential sweep. Interestingly, while the spectral gap of the $n$-th power of the down-up walk is still bounded from above by a constant, under a strong enough local spectral assumption (in the sense of Gur, Lifschitz, Liu, STOC 2022) we can show that the spectral gap of this walk can be arbitrarily close to 1. We also study other isoperimetric inequalities for these walks, and show that under the assumptions of local entropy contraction (related to the considerations of Gur, Lifschitz, Liu), these walks satisfy an entropy contraction inequality.