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Polylogarithmic Sketches for Clustering

Authors Moses Charikar , Erik Waingarten

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Author Details

Moses Charikar
  • Stanford University, CA, USA
Erik Waingarten
  • Stanford University, CA, USA

Funding

  • Charikar, Moses: Supported by a Simons Investigator Award.
  • Waingarten, Erik: Supported by the National Science Foundation under Award no. 2002201.

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Moses Charikar and Erik Waingarten. Polylogarithmic Sketches for Clustering. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 38:1-38:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022) https://doi.org/10.4230/LIPIcs.ICALP.2022.38

Abstract

Given n points in 𝓁_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-th powers of distances of points to their cluster centers. For p ∈ [1,2], we design sketches of size poly(log(nd),k,1/ε) such that the cost of the optimal clustering can be estimated to within factor 1+ε, 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/ε). 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 ∈ [1,2).

Subject Classification

ACM Subject Classification
  • Theory of computation → Sketching and sampling
Keywords
  • sketching
  • clustering

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