Distributed k-Means with Outliers in General Metrics

Abstract

Center-based clustering is a pivotal primitive for unsupervised learning and data analysis. A popular variant is the k-means problem, which, given a set P of points from a metric space and a parameter $k<|P|$, requires finding a subset $S \subset P$ of k points, dubbed centers, which minimizes the sum of all squared distances of points in P from their closest center. A more general formulation, 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 polynomial-time sequential (possibly bicriteria) approximation 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 its doubling dimension D. To the best of our knowledge, no previous distributed approaches were able to attain similar quality-performance tradeoffs for general metrics.