The Paulsen problem, continuous operator scaling, and smoothed analysis
The Paulsen problem, continuous operator scaling, and smoothed analysis
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Paulsen 问题、连续算子缩放和平滑分析
DOI：
10.1145/3188745.3188794
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
Akshay Ramachandran
中科院分区：
文献类型：
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作者：
T. C. Kwok;L. Lau;Y. Lee;Akshay Ramachandran
Paulsen 问题是算子理论中的一个基本开放问题：给定向量 u1, …, un ε ℝd，它们 є-几乎满足 Parseval 条件和等范条件，它是否接近完全满足 Parseval 条件和等范条件的一组向量 v1, …, vn ∈ ℝd？给定 u1, …, un，平方距离（到精确解集）定义为 infv Σi=1n || ui − vi ||22 其中下确界位于精确解集上。先前的结果表明，任何 є 近似解的平方距离至多为 O(poly(d,n,є))，并且存在平方距离至少为 Ω(d є) 的 є 近似解。基本的悬而未决的问题是平方距离是否可以独立于向量的数量 n。我们通过证明任何 є-近解的平方距离为 O(d13/2 є) 来肯定地回答这个问题。我们的方法基于算子缩放算法的连续版本，由两部分组成。首先，我们定义一个基于算子缩放的动力系统，并用它来证明任何 є-近解的平方距离为 O(d2 n є)。然后，我们表明，通过随机扰动输入向量，当 n 足够大且 є 足够小时，动态系统将收敛得更快，并且 є 近解的平方距离为 O(d5/2 є)。为了分析动力系统的收敛性，我们开发了一些限制算子容量的新技术，这是 Gurvits 引入的用于分析算子缩放算法的概念。
The Paulsen problem is a basic open problem in operator theory: Given vectors u1, …, un ∈ ℝd that are є-nearly satisfying the Parseval’s condition and the equal norm condition, is it close to a set of vectors v1, …, vn ∈ ℝd that exactly satisfy the Parseval’s condition and the equal norm condition? Given u1, …, un, the squared distance (to the set of exact solutions) is defined as infv ∑i=1n || ui − vi ||22 where the infimum is over the set of exact solutions. Previous results show that the squared distance of any є-nearly solution is at most O(poly(d,n,є)) and there are є-nearly solutions with squared distance at least Ω(d є). 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 є-nearly solution is O(d13/2 є). 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 є-nearly solution is O(d2 n є). Then, we show that by randomly perturbing the input vectors, the dynamical system will converge faster and the squared distance of an є-nearly solution is O(d5/2 є) when n is large enough and є 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.