The Paulsen Problem, Continuous Operator Scaling, and Smoothed Analysis

The Paulsen problem is a basic open problem in operator theory: Given vectors u(1) , . . . , u(n) is an element of R-d that are epsilon-nearly satisfying the Parseval's condition and the equal norm condition, is it close to a set of vectors v(1), . . . , v(n) is an element of R-d that exactly satisfy the Parseval's condition and the equal norm condition? Given u(1), . . . , u(n), the squared distance (to the set of exact solutions) is defined as inf(v) Sigma(n)(i=1) parallel to u(i) - v(i)parallel to(2)(2) where the infimum is over the set of exact solutions. Previous results show that the squared distance of any epsilon-nearly solution is at most O(poly(d, n, epsilon)) and there are epsilon-nearly solutions with squared distance at least O(d epsilon). 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 epsilon-nearly solution is O(d(13/2)epsilon). 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 epsilon-nearly solution is O(d(2)n epsilon). Then, we show that by randomly perturbing the input vectors, the dynamical system will converge faster and the squared distance of an epsilon-nearly solution is O(d(5/2)epsilon) when n is large enough and epsilon is small enough. To analyze the convergence of the dynamical system, we develop some newtechniques in lower bounding the operator capacity, a concept introduced by Gurvits to analyzing the operator scaling algorithm.