Distributed k-Means with Outliers in General Metrics

• URL: http://arxiv.org/abs/2202.08173v2
• Date: Fri, 18 Feb 2022 16:56:37 GMT
• Title: Distributed k-Means with Outliers in General Metrics
• Authors: Enrico Dandolo, Andrea Pietracaprina, Geppino Pucci
• Abstract summary: We present a distributed coreset-based 3-round approximation algorithm for k-means with $z$ outliers for general metric spaces. An important feature of our algorithm is that it obliviously adapts to the intrinsic complexity of the dataset, captured by the dimension doubling $D$ of the metric space.
• Score: 0.6117371161379208
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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