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

TL;DR

This paper introduces a novel polynomial-time algorithm for computing the noncommutative rank of linear symbolic matrices over any field, utilizing discrete convex optimization on CAT(0) spaces and submodular optimization on modular lattices.

Contribution

It presents a new polynomial-time algorithm that combines submodular and convex optimization techniques on CAT(0) spaces for nc-rank computation over arbitrary fields.

Findings

Algorithm works over any field nd is significantly different from previous methods.

Uses a novel combination of submodular and convex optimization techniques.

Achieves polynomial-time complexity for nc-rank computation.

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.