Operator Scaling via Geodesically Convex Optimization, Invariant Theory and Polynomial Identity Testing

TL;DR

This paper introduces a second-order geodesically convex optimization method for operator scaling, leading to polynomial-time algorithms for operator equivalence and a new class of Polynomial Identity Testing problems.

Contribution

It develops a novel second-order geodesically convex optimization approach for operator scaling, enabling polynomial-time solutions and advancing Polynomial Identity Testing.

Findings

Polynomial-time algorithm for operator scaling

Deterministic polynomial-time solution for new PIT class

Exponential improvement over previous methods

Abstract

We propose a new second-order method for geodesically convex optimization on the natural hyperbolic metric over positive definite matrices. We apply it to solve the operator scaling problem in time polynomial in the input size and logarithmic in the error. This is an exponential improvement over previous algorithms which were analyzed in the usual Euclidean, "commutative" metric (for which the above problem is not convex). Our method is general and applicable to other settings. As a consequence, we solve the equivalence problem for the left-right group action underlying the operator scaling problem. This yields a deterministic polynomial-time algorithm for a new class of Polynomial Identity Testing (PIT) problems, which was the original motivation for studying operator scaling.