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Motivated by recent Linear Programming solvers, we design dynamic datastructures 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-Morrisonidentity for implementing $\textit{cascading}$ low-rank updates, combined withrecent sketching technology. Our techniques and amortized analysis ofmulti-level partial updates, may be of broader interest to dynamic matrixproblems.  This data structure leads to the fastest known LP solver for general (dense)linear programs, improving the running time of the recent algorithms of (Cohenet al.'19, Lee et al.'19, Brand'20) from $O^*(n^{2+ \max\{\frac{1}{6},\omega-2, \frac{1-\alpha}{2}\}})$ to $O^*(n^{2+\max\{\frac{1}{18}, \omega-2,\frac{1-\alpha}{2}\}})$, where $\omega$ and $\alpha$ are the fast matrixmultiplication exponent and its dual. Hence, under the common belief that$\omega \approx 2$ and $\alpha \approx 1$, our LP solver runs in$O^*(n^{2.055})$ time instead of $O^*(n^{2.16})$.

Faster Dynamic Matrix Inverse for Faster LPs