This paper presents an algorithm for Hermitian diagonalization running in near matrix multiplication time requiring only 2lg(1/ε ) + O (log(n ) + log log(1/ε )) bits of precision. Despite the widespread, highly successful use of various algorithms for Hermitian diagonalization in practice, the literature long lacked rigorous guarantees of their performance in finite arithmetic. The recent work of Banks, Garza-Vargas, Kulkarni, and Srivastava (FOCS 2020) changed this by providing an algorithm for diagonalizing any matrix up to backward error, and proving it requires no more than O (log4(n/ε ) log(n )) bits. This work improves upon their algorithm in the Hermitian setting to dramatically reduce the bit requirement.

We prove upper bounds on the graph diameters of polytopes in two settings. The first is a worst-case bound for integer polytopes in terms of the length of the description of the polytope (in bits) and the minimum angle between facets of its polar. The second is a smoothed analysis bound: given an appropriately normalized polytope, we add small Gaussian noise to each constraint. We consider a natural geometric measure on the vertices of the perturbed polytope (corresponding to the mean curvature measure of its polar) and show that with high probability there exists a "giant component" of vertices, with measure 1-o(1) and polynomial diameter. Both bounds rely on spectral gaps - of a certain Schrödinger operator in the first case, and a certain continuous time Markov chain in the second - which arise from the log-concavity of the volume of a simple polytope in terms of its slack variables.

Let $M$ be an arbitrary $n$ by $n$ matrix of rank $n-k$. We study the condition number of $M$ plus a \emph{low-rank} perturbation $UV^T$ where $U, V$ are $n$ by $k$ random Gaussian matrices. Under some necessary assumptions, it is shown that $M+UV^T$ is unlikely to have a large condition number. The main advantages of this kind of perturbation over the well-studied dense Gaussian perturbation, where every entry is independently perturbed, is the $O(nk)$ cost to store $U,V$ and the $O(nk)$ increase in time complexity for performing the matrix-vector multiplication $(M+UV^T)x$. This improves the $Ω(n^2)$ space and time complexity increase required by a dense perturbation, which is especially burdensome if $M$ is originally sparse. Our results also extend to the case where $U$ and $V$ have rank larger than $k$ and to symmetric and complex settings. We also give an application to linear systems solving and perform some numerical experiments. Lastly, barriers in applying low-rank noise to other problems studied in the smoothed analysis framework are discussed.