Tutte Colloquium - Tsz Chiu Kwok

Title: The Paulsen problem, continuous operator scaling, and smoothed analysis

Abstract:

The Paulsen problem is a basic open problem in operator theory: Given vectors u_1, ..., u_n in R^d that are eps-nearly satisfying the Parseval's condition and the equal norm condition, is it close to a set of vectors v_1, ..., v_n in 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 eps-nearly solution is at most O(poly(d,n,eps)) and there are eps-nearly solutions with squared distance at least Omega(d eps). 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 eps-nearly solution is O(d^13/2eps). Our approach is based on a continuous version of the operator scaling algorithm and consists of two parts. First, we dene a dynamical system based on operator scaling and use it to prove that the squared distance of any eps-nearly solution is O(d2neps).
Then, we show that by randomly perturbing the input vectors, the  dynamical system will converge faster and the squared distance of an eps-nearly solution is O(d^5/2eps) when n is large enough and eps 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 analyzing the operator scaling algorithm.
Joint work with Lap Chi Lau, Yin Tat Lee, and Akshay Ramachandran.