The Paulsen problem, continuous operator scaling, and smoothed analysis

Abstract

The Paulsen problem is a basic open problem in operator theory. We define a continuous version of the operator scaling algorithm to solve this problem. A key step is to show that the continuous operator scaling algorithm converges faster in a perturbed input. To this end, we develop some new techniques in lower bounding the operator capacity, a concept introduced by Gurvits to analyze the operator scaling algorithm. The talk will be self-contained. Joint work with Tsz Chiu Kwok, Yin Tat Lee, and Akshay Ramachandran.