Randomized Dimensionality Reduction for Euclidean Maximization and Diversity Measures

2025-05-30

AI Summary

This paper investigates the impact of random dimensionality reduction on several maximization problems in Euclidean space—namely, maximum matching, maximum spanning tree, and maximum traveling salesman problem—as well as on dataset diversity measures. It introduces a novel analytical framework centered on the dataset’s *doubling dimension* λ_X, establishing that O(λ_X) random projection dimensions suffice to preserve optimal objective values within a (1±ε)-multiplicative factor—improving upon classical bounds dependent on the number of points |X|. The derived lower bound is shown to be tight. Theoretical analysis guarantees near-preserving solution quality, while empirical evaluation confirms high post-projection accuracy and substantial computational speedup. The core contribution is the first quantitative characterization linking dimensionality reduction efficacy directly to the intrinsic geometric complexity λ_X, yielding a finer-grained and more practically relevant theoretical foundation for optimization in high-dimensional spaces.

Technology Category

• Machine Learning: Dimensionality Reduction/Feature Selection
• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

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.

Problem

• Studies dimension reduction for Euclidean maximization problems
• Links reduction effectiveness to dataset's doubling dimension
• Proves target dimension preserves near-optimal solution values

Innovation

• Randomized dimensionality reduction for Euclidean maximization
• Target dimension O(λ_X) preserves near-optimal solutions
• Empirical validation of solution quality and speedups