VAC: A Volume-sampling-based Elimination Rule for Approximate Cholesky Factorization

2026-07-27

AI Summary

This study addresses the issues of excessive fill-in in approximate Cholesky factorization and the lack of theoretical guarantees and graph connectivity in existing sampling rules. To overcome these limitations, this work proposes a volume-sampling-based elimination strategy that optimizes the elimination ordering by uniformly and randomly generating maximum spanning trees of product cliques. This approach effectively integrates the theoretical correctness of Kyng’s framework with the practical connectivity of Gao’s method, strictly preserving marginal edge distributions while ensuring graph connectivity. By combining linear time complexity with O(log n)-depth parallel computing techniques, the proposed algorithm yields a concise, provably correct, and efficiently parallelizable scheme for approximate Cholesky factorization.

Abstract

We propose Volume Appproximate Cholesky (VAC), an alternative sampling rule for practical approximate Cholesky algorithms. Our rule samples a uniformly random spanning tree of the arising product clique to reduce the fill-in generated at each step. Sampling a random spanning tree preserves the edgewise marginals of the provably correct scheme of (Kyng \&Sachdeva 2016), while ensuring connectivity in the spirit of the practical rule proposed in (Gao, Kyng \&Spielman 2023). Our sampling method is simple, provably linear time and also admits a $O(\log n)$ depth parallel implementation.

Problem

• Approximate Cholesky Factorization
• Fill-in Reduction
• Sampling Rule
• Spanning Tree

Innovation

• Approximate Cholesky Factorization
• Volume Sampling
• Random Spanning Tree
• Fill-in Reduction
• Parallel Implementation