  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 $O^*(n^{2+ \max\{\frac{1}{6},
\omega-2, \frac{1-\alpha}{2}\}})$ to $O^*(n^{2+\max\{\frac{1}{18}, \omega-2,
\frac{1-\alpha}{2}\}})$, where $\omega$ and $\alpha$ are the fast matrix
multiplication 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 $O^*(n^{2.16})$.
