Computing the nc-rank via discrete convex optimization on CAT(0) spaces

Abstract

In this paper, we address the noncommutative rank (nc-rank) computation of a linear symbolic matrix \[ A = A_1 x_1 + A_2 x_2 + \cdots + A_m x_m, \] where each $A_i$ is an $n \times n$ matrix over a field $\mathbb{K}$, and $x_i$ $(i=1,2,\ldots,m)$ are noncommutative variables. For this problem, polynomial time algorithms were given by Garg, Gurvits, Oliveira, and Wigderson for $\mathbb{K} = \mathbb{Q}$, and by Ivanyos, Qiao, and Subrahmanyam for an arbitrary field $\mathbb{K}$. We present a significantly different polynomial time algorithm that works on an arbitrary field $\mathbb{K}$. Our algorithm is based on a combination of submodular optimization on modular lattices and convex optimization on CAT(0) spaces.