Operator Scaling: Theory and Applications

In this paper, we present a deterministic polynomial time algorithm for testing whether a symbolic matrix in non-commuting variables over $${\mathbb {Q}}$$ Q is invertible or not. The analogous question for commuting variables is the celebrated polynomial identity testing (PIT) for symbolic determinants. In contrast to the commutative case, which has an efficient probabilistic algorithm, the best previous algorithm for the non-commutative setting required exponential time (Ivanyos et al. in Comput Complex 26(3):717–763, 2017 ) (whether or not randomization is allowed). The algorithm efficiently solves the “word problem” for the free skew field, and the identity testing problem for arithmetic formulae with division over non-commuting variables, two problems which had only exponential time algorithms prior to this work. The main contribution of this paper is a complexity analysis of an existing algorithm due to Gurvits (J Comput Syst Sci 69(3):448–484, 2004 ), who proved it was polynomial time for certain classes of inputs. We prove it always runs in polynomial time. The main component of our analysis is a simple (given the necessary known tools) lower bound on central notion of capacity of operators (introduced by Gurvits  2004 ). We extend the algorithm to actually approximate capacity to any accuracy in polynomial time, and use this analysis to give quantitative bounds on the continuity of capacity (the latter is used in a subsequent paper on Brascamp–Lieb inequalities). We also extend the algorithm to compute not only singularity, but actually the (non-commutative) rank of a symbolic matrix, yielding a factor 2 approximation of the commutative rank. This naturally raises a relaxation of the commutative PIT problem to achieving better deterministic approximation of the commutative rank. Symbolic matrices in non-commuting variables, and the related structural and algorithmic questions, have a remarkable number of diverse origins and motivations. They arise independently in (commutative) invariant theory and representation theory, linear algebra, optimization, linear system theory, quantum information theory, approximation of the permanent and naturally in non-commutative algebra. We provide a detailed account of some of these sources and their interconnections. In particular, we explain how some of these sources played an important role in the development of Gurvits’ algorithm and in our analysis of it here.