Operator Scaling: Theory and Applications

Abstract

In this paper we present a deterministic polynomial time algorithm for testing if a symbolic matrix in non-commuting variables over 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 [IQS17] (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 noncommuting 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 [Gur04], 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 [Gur04]). 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. * Microsoft Research New England, email: garga@microsoft.com.

Spectral Analysis of Matrix Scaling and Operator Scaling

“…In this algorithm, we alternately find a left scaling matrix L = ( i A i A * i ) −1/2 and set A i ← LA i so that the first condition of doubly balanced is satisfied, and a right scaling matrix R = ( i A * i A i ) −1/2 and set A i ← A i R so that the second condition of doubly balanced is satisfied, and repeat. This alternating algorithm is partially analyzed in [29] and is fully analyzed in [20,19]. Theorem 1.2 ([54,18,20,19]).…”

“…This alternating algorithm is partially analyzed in [29] and is fully analyzed in [20,19]. Theorem 1.2 ([54,18,20,19]). The alternating scaling algorithm returns an η-nearly doubly balanced scaling in O(poly(n, m, k, 1/η)) iterations if such a scaling exists.…”

“…This theorem is used in [20,19] to give the first polynomial time algorithm for computing the non-commutative rank of a symbolic matrix, as it is sufficient to set η to be inverse polynomial in n to solve that problem exactly. For some applications, however, faster convergence of η is required.…”

“…The operator scaling problem was used to the Brascamp-Lieb constant [21] and to compute the non-commutative rank of a symbolic matrix [20]. It is also used in [1] to solve the orbit intersection problem for the left-right group action.…”

“…In [20,19,29], a polynomial time algorithm for computing the non-commutative rank of a symbolic matrix is designed using operator scaling. Given A = (A 1 , .…”