Pipage Rounding, Pessimistic Estimators and Matrix Concentration

TL;DR

This paper introduces a new technique called concavity of pessimistic estimators to extend concentration results for sums of matrices and submodular functions under pipage rounding, with applications to spectral graph theory and optimization.

Contribution

It presents a novel method for proving concentration inequalities in negatively dependent settings, including a new variant of Lieb's theorem, and demonstrates multiple applications.

Findings

Concentration of matrix sums under pipage rounding is established.

A polynomial-time algorithm for constructing spectrally-thin trees is developed.

Applications include improved rounding techniques for semidefinite programs and spectral graph problems.

Abstract

Pipage rounding is a dependent random sampling technique that has several interesting properties and diverse applications. One property that has been particularly useful is negative correlation of the resulting vector. Unfortunately negative correlation has its limitations, and there are some further desirable properties that do not seem to follow from existing techniques. In particular, recent concentration results for sums of independent random matrices are not known to extend to a negatively dependent setting. We introduce a simple but useful technique called concavity of pessimistic estimators. This technique allows us to show concentration of submodular functions and concentration of matrix sums under pipage rounding. The former result answers a question of Chekuri et al. (2009). To prove the latter result, we derive a new variant of Lieb's celebrated concavity theorem in matrix analysis. We provide numerous applications of these results. One is to spectrally-thin trees, a spectral analog of the thin trees that played a crucial role in the recent breakthrough on the asymmetric traveling salesman problem. We show a polynomial time algorithm that, given a graph where every edge has effective conductance at least $\kappa$, returns an $O(\kappa^{-1} \cdot \log n / \log \log n)$-spectrally-thin tree. There are further applications to rounding of semidefinite programs, to the column subset selection problem, and to a geometric question of extracting a nearly-orthonormal basis from an isotropic distribution.