Randomized Dimensionality Reduction for Euclidean Maximization and Diversity Measures

Jie Gao; Rajesh Jayaram; Benedikt Kolbe; Shay Sapir; Chris Schwiegelshohn; Sandeep Silwal; Erik Waingarten

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arXiv · 2506.00165

Randomized Dimensionality Reduction for Euclidean Maximization and Diversity Measures

Jie Gao·Rajesh Jayaram·Benedikt Kolbe·Shay Sapir·Chris Schwiegelshohn·Sandeep Silwal·Erik Waingarten

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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.

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Jie Gao, Rajesh Jayaram, Benedikt Kolbe, Shay Sapir, Chris Schwiegelshohn, Sandeep Silwal, Erik Waingarten. 2025-05-30. Randomized Dimensionality Reduction for Euclidean Maximization and Diversity Measures. https://arxiv.org/abs/2506.00165

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