Random zero sets with local growth guarantees

2024-10-29

arXiv.org

AI Summary

This paper addresses the construction of random zero sets in $n$-point metric spaces satisfying a local growth condition, aiming to ensure that for any pair of points at distance $geq au$, one lies in the zero set while the other remains at bounded distance from it—achieving a constant lower bound on this separation probability. Methodologically, it refines the ARV rounding technique for zero-set construction under metric embeddings, establishing for the first time a quantitative link between local ball-volume ratios and separation probability. By integrating quasisymmetric embedding analysis with probabilistic metric geometry, the work derives that the maximum Euclidean distortion of any $n$-point set embeddable into $ell_1$ is $Theta(sqrt{log n})$. Consequently, it rigorously confirms that the integrality gap of the Goemans–Linial semidefinite program for the sparsest cut problem is also $Theta(sqrt{log n})$.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We prove that if $(mathcal{M},d)$ is an $n$-point metric space that embeds quasisymmetrically into a Hilbert space, then for every $ au>0$ there is a random subset $mathcal{Z}$ of $mathcal{M}$ such that for any pair of points $x,yin mathcal{M}$ with $d(x,y)ge au$, the probability that both $xin mathcal{Z}$ and $d(y,mathcal{Z})ge eta au/sqrt{1+log (|B(y,kappa eta au)|/|B(y,eta au)|)}$ is $Omega(1)$, where $kappa>1$ is a universal constant and $eta>0$ depends only on the modulus of the quasisymmetric embedding. The proof relies on a refinement of the Arora--Rao--Vazirani rounding technique. Among the applications of this result is that the largest possible Euclidean distortion of an $n$-point subset of $ell_1$ is $Theta(sqrt{log n})$, and the integrality gap of the Goemans--Linial semidefinite program for the Sparsest Cut problem on inputs of size $n$ is $Theta(sqrt{log n})$. Multiple further applications are given.

Problem

• Study random subsets in quasisymmetric Hilbert embeddings
• Analyze Euclidean distortion in n-point subsets of l1
• Evaluate integrality gaps in Sparsest Cut SDPs

Innovation

• Random subset construction with growth guarantees
• Refined Arora-Rao-Vazirani rounding technique
• Quasisymmetric embedding into Hilbert space