Functional Analysis Seminar: The Paulsen problem, continuous operator scaling, and smoothed analysis

Seminar | December 1 | 2:10-3 p.m. | 748 Evans Hall | Note change in location

 Tsz Chiu Kwok, University of Waterloo

 Department of Mathematics

The Paulsen problem is a basic open problem in operator theory: Given vectors $u_1, ..., u_n$ in $\mathbb R^d$ that are $\varepsilon $-nearly satisfying the Parseval's condition and the equal norm condition, is it close to a set of vectors $v_1, ..., v_n$ in $\mathbb R^d$ that exactly satisfy the Parseval's condition and the equal norm condition? Given $u_1,..., u_n$, we consider the squared distance to the set of exact solutions. Previous results show that the squared distance of any $\varepsilon $-nearly solution is at most $O(poly(d, n, \varepsilon ))$ and there are $\varepsilon $-nearly solutions with squared distance at least $Omega(d \varepsilon )$. The fundamental open question is whether the squared distance can be independent of the number of vectors n.

We answer this question affirmatively by proving that the squared distance of any $\varepsilon $-nearly solution is $O(d^{13/2} \varepsilon )$. Our approach is based on a continuous version of the operator scaling algorithm and consists of two parts. First, we define a dynamical system based on operator scaling and use it to prove that the squared distance of any eps-nearly solution is $O(d^2 n \varepsilon )$. Then, we show that by randomly perturbing the input vectors, the dynamical system will converge faster and the squared distance of an $\varepsilon $-nearly solution is $O(d^{5/2} \varepsilon )$ when n is large enough and $\varepsilon $ is small enough. To analyze the convergence of the dynamical system, we develop some new techniques in lower bounding the operator capacity, a concept introduced by Gurvits to analyze the operator scaling algorithm.

Joint work with Lap Chi Lau, Yin Tat Lee, and Akshay Ramachandran.