2020/04/16 by Shunhua Jiang, Jiang, Shunhua, Zhao Song +5 · 55 citations
Computer Science · Engineering · Mathematics · #Approx #Combinatorics #Complexity and Algorithms in Graphs #Computer science #Data Structures and Algorithms (cs.DS) #Discrete mathematics #Exponent #FOS: Computer and information sciences #Geometry #Inverse #Mathematical optimization #Mathematics #Matrix (chemical analysis) #Matrix multiplication #Omega #Physics #Rank (graph theory) #Solver #Sparse and Compressive Sensing Techniques #Stochastic Gradient Optimization Techniques #cs.DS
paper · pdf · doi:10.48550/arxiv.2004.07470
published in arXiv (Cornell University) (Cornell University)
arxiv created 2020/04/16 · openalex publication_date 2020/04/16 · arxiv updated 2020/04/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/08
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).