Some problems in asymptotic convex geometry and random matrices motivated by numerical algorithms
Roman Vershynin
TL;DR
The paper investigates problems in asymptotic convex geometry motivated by the simplex method in Linear Programming, focusing on the size of planar sections of high-dimensional polytopes and on the invertibility of random matrices. It develops a dual framework based on projections and polar duality, linking the shadow-vertex pivot walk to the edge structure of the section $K \cap E$ with $K = P^\circ$. A main result under smoothed analysis is that the expected number of edges of $K \cap E$ is polylogarithmic in the ambient dimension, with a concrete bound for Gaussian perturbations: $ \mathbb{E} |\mathrm{edges}(K \cap E)| \le C d^3 \sigma^{-4}$. On the random-matrix side, the paper surveys sharp nonasymptotic and asymptotic bounds for the singular values of Gaussian and subgaussian matrices, including that the smallest singular value of a square matrix is of order $n^{-1/2}$ and is controlled with high probability. Together these results underpin polynomial smoothed complexity of shadow-vertex methods and yield robust invertibility guarantees that inform nondegeneracy assumptions in convex-geometric analyses.
Abstract
The simplex method in Linear Programming motivates several problems of asymptotic convex geometry. We discuss some conjectures and known results in two related directions -- computing the size of projections of high dimensional polytopes and estimating the norms of random matrices and their inverses.