2020/04/16 by Shunhua Jiang, Jiang, Shunhua, Zhao Song +5 · 4 citations
Computer Science · Engineering · #Complexity and Algorithms in Graphs #Data Structures and Algorithms (cs.DS) #FOS: Computer and information sciences #Sparse and Compressive Sensing Techniques #Stochastic Gradient Optimization Techniques
paper · pdf · doi:10.48550/arxiv.2004.07470
openalex publication_date 2020/04/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by recent Linear Programming solvers, we design dynamic data structures for maintaining the inverse of an n× n real matrix under low-rank updates, with polynomially faster amortized running time. Our data structure is based on a recursive application of the Woodbury-Morrison identity for implementing cascading low-rank updates, combined with recent sketching technology. Our techniques and amortized analysis of multi-level partial updates, may be of broader interest to dynamic matrix problems. This data structure leads to the fastest known LP solver for general (dense) linear programs, improving the running time of the recent algorithms of (Cohen et al.'19, Lee et al.'19, Brand'20) from O^*(n^2+ max\(1)/(6), ω-2, (1-α)/(2)\) to O^*(n^2+max\(1)/(18), ω-2, (1-α)/(2)\), where ω and α are the fast matrix multiplication exponent and its dual. Hence, under the common belief that ω≈ 2 and α≈ 1, our LP solver runs in O^*(n2.055) time instead of O^*(n2.16).