论文标题
更快的动态矩阵逆，用于更快的LPS
Faster Dynamic Matrix Inverse for Faster LPs
论文作者
论文摘要
在最近的线性编程求解器的激励下，我们设计了动态数据结构，以维持$ n \ times n $ real矩阵的倒数，下面是$ \ textit {low-Rank} $更新，并且多发摊销的运行时间更快。我们的数据结构是基于伍德伯里 - 莫里森身份的递归应用，用于实现$ \ textit {cascading} $低级更新，并结合最近的草图技术。我们对多层次部分更新的技术和摊销分析，对于动态矩阵问题可能会更广泛。 这种数据结构导致了通用（密度）线性程序的最快的LP求解器，从而改善了（Cohen et al.'19，Lee et al.'19，Brand'20）的最新算法的运行时间，从$ o^*（n^{2+ \ max \ max \ max \ frac） \frac{1-α}{2}\}})$ to $O^*(n^{2+\max\{\frac{1}{18}, ω-2, \frac{1-α}{2}\}})$, where $ω$ and $α$ are the fast matrix multiplication exponent and its dual.因此，在人们普遍认为$ω\大约2 $和$α\大约1 $的情况下，我们的LP求解器以$ o^*（n^{2.055}）$时间运行，而不是$ o^*（n^{2.16}）$。
Motivated by recent Linear Programming solvers, we design dynamic data structures for maintaining the inverse of an $n\times n$ real matrix under $\textit{low-rank}$ updates, with polynomially faster amortized running time. Our data structure is based on a recursive application of the Woodbury-Morrison identity for implementing $\textit{cascading}$ 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 $O^*(n^{2+ \max\{\frac{1}{6}, ω-2, \frac{1-α}{2}\}})$ to $O^*(n^{2+\max\{\frac{1}{18}, ω-2, \frac{1-α}{2}\}})$, where $ω$ and $α$ are the fast matrix multiplication exponent and its dual. Hence, under the common belief that $ω\approx 2$ and $α\approx 1$, our LP solver runs in $O^*(n^{2.055})$ time instead of $O^*(n^{2.16})$.
