Dimension Reduction for Clustering: The Curious Case of Discrete Centers

2025-09-09

AI Summary

This work studies dimensionality reduction for discrete center clustering—where cluster centers must be selected from the input dataset. Addressing the inadequacy of classical Johnson–Lindenstrauss embeddings under discrete center constraints, we introduce the doubling dimension to this setting and propose a cost-preserving metric embedding. We prove that $Oig(varepsilon^{-2}(ddim + log k + loglog n)ig)$ dimensions suffice to $(1pmvarepsilon)$-approximate the cost of any $k$-center clustering. By relaxing the stability condition on optimal solutions, we eliminate the $loglog n$ term, achieving an almost-optimal dimension bound of $Oig(varepsilon^{-2}(ddim + log k)ig)$. Our results extend classical dimensionality reduction theory to discrete center clustering, providing tight, constructive guarantees for compressing high-dimensional data while preserving clustering cost structure—enabling efficient and theoretically sound discrete $k$-center clustering in reduced dimensions.

Technology Category

• Machine Learning: Clustering
• Data Mining & Knowledge Management: Data Compression
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: LLM based quality controls for crowd work

Abstract

The Johnson-Lindenstrauss transform is a fundamental method for dimension reduction in Euclidean spaces, that can map any dataset of $n$ points into dimension $O(log n)$ with low distortion of their distances. This dimension bound is tight in general, but one can bypass it for specific problems. Indeed, tremendous progress has been made for clustering problems, especially in the emph{continuous} setting where centers can be picked from the ambient space $mathbb{R}^d$. Most notably, for $k$-median and $k$-means, the dimension bound was improved to $O(log k)$ [Makarychev, Makarychev and Razenshteyn, STOC 2019]. We explore dimension reduction for clustering in the emph{discrete} setting, where centers can only be picked from the dataset, and present two results that are both parameterized by the doubling dimension of the dataset, denoted as $operatorname{ddim}$. The first result shows that dimension $O_ε(operatorname{ddim} + log k + loglog n)$ suffices, and is moreover tight, to guarantee that the cost is preserved within factor $1pmε$ for every set of centers. Our second result eliminates the $loglog n$ term in the dimension through a relaxation of the guarantee (namely, preserving the cost only for all approximately-optimal sets of centers), which maintains its usefulness for downstream applications. Overall, we achieve strong dimension reduction in the discrete setting, and find that it differs from the continuous setting not only in the dimension bound, which depends on the doubling dimension, but also in the guarantees beyond preserving the optimal value, such as which clusterings are preserved.

Problem

• Dimension reduction for discrete clustering with centers from dataset
• Tight dimension bound based on doubling dimension and k
• Eliminating log log n term by relaxing cost guarantees

Innovation

• Dimension reduction using doubling dimension parameter
• Tight bound with O_ε(ddim + log k + log log n)
• Relaxed guarantee eliminates log log n term