In this paper, we address the noncommutative rank (nc-rank) computation of a linear symbolic matrix A=A1 x1+A2 x2 +⋯+ Am xm, where each Ai is an n x 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.

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