Distributed k-Means with Outliers in General Metrics

TL;DR

This paper introduces a distributed coreset-based 3-round approximation algorithm for k-means with outliers in general metric spaces, achieving near-optimal solutions with low memory requirements and adaptability to dataset complexity.

Contribution

It presents the first distributed algorithm for k-means with outliers in general metrics that balances solution quality, memory efficiency, and dataset complexity adaptation.

Findings

Requires sublinear local memory per reducer.

Achieves an approximation ratio close to the best sequential algorithms.

Adapts to dataset complexity via doubling dimension D.

Abstract

Center-based clustering is a pivotal primitive for unsupervised learning and data analysis. A popular variant is undoubtedly the k-means problem, which, given a set $P$ of points from a metric space and a parameter $k<|P|$, requires to determine a subset $S$ of $k$ centers minimizing the sum of all squared distances of points in $P$ from their closest center. A more general formulation, known as k-means with $z$ outliers, introduced to deal with noisy datasets, features a further parameter $z$ and allows up to $z$ points of $P$ (outliers) to be disregarded when computing the aforementioned sum. We present a distributed coreset-based 3-round approximation algorithm for k-means with $z$ outliers for general metric spaces, using MapReduce as a computational model. Our distributed algorithm requires sublinear local memory per reducer, and yields a solution whose approximation ratio is an additive term $O(\gamma)$ away from the one achievable by the best known sequential (possibly bicriteria) algorithm, where $\gamma$ can be made arbitrarily small. An important feature of our algorithm is that it obliviously adapts to the intrinsic complexity of the dataset, captured by the doubling dimension $D$ of the metric space. To the best of our knowledge, no previous distributed approaches were able to attain similar quality-performance tradeoffs for general metrics.