Polylogarithmic Sketches for Clustering

TL;DR

This paper introduces polylogarithmic-sized sketches for efficiently estimating clustering costs in high-dimensional spaces, enabling streaming and distributed algorithms with sublinear dependence on dimension for p in [1,2].

Contribution

The authors develop novel sketches that approximate clustering costs in high-dimensional spaces with polylogarithmic size, applicable to streaming and distributed settings, for p in [1,2].

Findings

Sketch size is poly$( ext{log}(nd),k,1/ extepsilon)$.

Provides the first sublinear dependence on $d$ for $p ext{ in } [1,2)$ in streaming and distributed algorithms.

Achieves $(1+ extepsilon)$-approximation of clustering cost without recovering cluster centers.

Abstract

Given $n$ points in $\ell_p^d$, we consider the problem of partitioning points into $k$ clusters with associated centers. The cost of a clustering is the sum of $p^{\text{th}}$ powers of distances of points to their cluster centers. For $p \in [1,2]$, we design sketches of size poly$(\log(nd),k,1/\epsilon)$ such that the cost of the optimal clustering can be estimated to within factor $1+\epsilon$, despite the fact that the compressed representation does not contain enough information to recover the cluster centers or the partition into clusters. This leads to a streaming algorithm for estimating the clustering cost with space poly$(\log(nd),k,1/\epsilon)$. We also obtain a distributed memory algorithm, where the $n$ points are arbitrarily partitioned amongst $m$ machines, each of which sends information to a central party who then computes an approximation of the clustering cost. Prior to this work, no such streaming or distributed-memory algorithm was known with sublinear dependence on $d$ for $p \in [1,2)$.