Polynomial-time approximation schemes for geometric min-sum median clustering

The Johnson--Lindenstrauss lemma states that <i>n</i> points in ahigh-dimensional Hilbert space can be embedded with smalldistortion of the distances into an <i>O</i>(log <i>n</i>)dimensional space by applying a random linear transformation. Weshow that similar (though weaker) properties hold for certainrandom linear transformations over the Hamming cube. We use thesetransformations to solve NP-hard clustering problems in the cube aswell as in geometric settings.More specifically, we address thefollowing clustering problem. Given <i>n</i> points in a larger set(e.g., ℝ^d) endowed with a distance function (e.g.,<i>L</i>² distance), we would like to partition the dataset into <i>k</i> disjoint clusters, each with a "cluster center,"so as to minimize the sum over all data points of the distancebetween the point and the center of the cluster containing thepoint. The problem is provably NP-hard in some high-dimensionalgeometric settings, even for <i>k</i> = 2. We give polynomial-timeapproximation schemes for this problem in several settings,including the binary cube {0,1}^d with Hamming distance,and ℝ^d either with <i>L</i>¹ distance,or with <i>L</i>² distance, or with the square of<i>L</i>² distance. In all these settings, the bestprevious results were constant factor approximation guarantees.Wenote that our problem is similar in flavor to the <i>k</i>-medianproblem (and the related facility location problem), which has beenconsidered in graph-theoretic and fixed dimensional geometricsettings, where it becomes hard when <i>k</i> is part of the input.In contrast, we study the problem when <i>k</i> is fixed, but thedimension is part of the input.