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The Paulsen problem, continuous operator scaling, and smoothed analysis

Banff International Research Station for Mathematical Innovation and Discovery

The Paulsen problem is a basic open problem in operator theory: Given vectors u_1, ..., u_n in R^d that are eps-close to 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-close solution is at most O(poly(d,n,eps)) and there are eps-close 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-close solution is O(d^7 eps). 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-close solution is O(d^2 n eps). Then, we show that by randomly perturbing the input vectors, the dynamical system will converge faster and the squared distance of an eps-close solution is O(d^3 eps) 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.

Mathematics; Computer science; Operations research, mathematical programming; Theoretical computer science

video/mp4

Author affiliation: University of Waterloo

2018-05-17

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/65965

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/