Computing the nc-Rank via Discrete Convex Optimization on CAT(0) Spaces

In this paper, we address the noncommutative rank (nc-rank) computation of a linear symbolic matrix A = A1x1 + A2x2 + · · ·+ Amxm, where each Ai is an n × n matrix over a field K, and xi (i = 1, 2, . . . ,m) are noncommutative variables. For this problem, polynomial time algorithms were given by Garg, Gurvits, Oliveira, and Wigderson for K = Q, and by Ivanyos, Qiao, and Subrahmanyam for an arbitrary field K. We present a significantly different polynomial time algorithm that works on an arbitrary field K. Our algorithm is based on a combination of submodular optimization on modular lattices and convex optimization on CAT(0) spaces.