The Noncommutative Edmonds' Problem Re-visited (2305.09984v1)

Abstract: Let $T$ be a matrix whose entries are linear forms over the noncommutative variables $x_1, x_2, \ldots, x_n$. The noncommutative Edmonds' problem (NSINGULAR) aims to determine whether $T$ is invertible in the free skew field generated by $x_1,x_2,\ldots,x_n$. Currently, there are three different deterministic polynomial-time algorithms to solve this problem: using operator scaling [Garg, Gurvits, Oliveira, and Wigserdon (2016)], algebraic methods [Ivanyos, Qiao, and Subrahmanyam (2018)], and convex optimization [Hamada and Hirai (2021)]. In this paper, we present a simpler algorithm for the NSINGULAR problem. While our algorithmic template is similar to the one in Ivanyos et. al.(2018), it significantly differs in its implementation of the rank increment step. Instead of computing the limit of a second Wong sequence, we reduce the problem to the polynomial identity testing (PIT) of noncommutative algebraic branching programs (ABPs). This enables us to bound the bit-complexity of the algorithm over $\mathbb{Q}$ without requiring special care. Moreover, the rank increment step can be implemented in quasipolynomial-time even without an explicit description of the coefficient matrices in $T$. This is possible by exploiting the connection with the black-box PIT of noncommutative ABPs [Forbes and Shpilka (2013)].