Randomized Dimensionality Reduction for Euclidean Maximization and Diversity Measures

TL;DR

This paper demonstrates that randomized dimensionality reduction preserves the quality of solutions for various Euclidean maximization and diversity problems, with the required dimension depending on the dataset's intrinsic doubling dimension.

Contribution

It establishes that a target dimension proportional to the dataset's doubling dimension suffices for approximate solution preservation, contrasting classical results that depend on dataset size.

Findings

Dimension reduction preserves near-optimal solutions effectively.

Target dimension of O(λ_X) is sufficient and necessary for certain problems.

Empirical results show solution quality and speedups in practice.

Abstract

Randomized dimensionality reduction is a widely-used algorithmic technique for speeding up large-scale Euclidean optimization problems. In this paper, we study dimension reduction for a variety of maximization problems, including max-matching, max-spanning tree, max TSP, as well as various measures for dataset diversity. For these problems, we show that the effect of dimension reduction is intimately tied to the \emph{doubling dimension} $\lambda_X$ of the underlying dataset $X$ -- a quantity measuring intrinsic dimensionality of point sets. Specifically, we prove that a target dimension of $O(\lambda_X)$ suffices to approximately preserve the value of any near-optimal solution,which we also show is necessary for some of these problems. This is in contrast to classical dimension reduction results, whose dependence increases with the dataset size $|X|$. We also provide empirical results validating the quality of solutions found in the projected space, as well as speedups due to dimensionality reduction.