A unified framework for approximating and clustering data

Dan Feldman, Michael Langberg

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

Given a set F of n positive functions over a ground set X, we consider the problem of computing x* that minimizes the expression Σ f ∈ Ff(x), over x ∈ X. A typical application is shape fitting, where we wish to approximate a set P of n elements (say, points) by a shape x from a (possibly infinite) family X of shapes. Here, each point p ∈ P corresponds to a function f such that f(x) is the distance from p to x, and we seek a shape x that minimizes the sum of distances from each point in P. In the k-clustering variant, each x\in X is a tuple of k shapes, and f(x) is the distance from p to its closest shape in x. Our main result is a unified framework for constructing coresets and approximate clustering for such general sets of functions. To achieve our results, we forge a link between the classic and well defined notion of ε-approximations from the theory of PAC Learning and VC dimension, to the relatively new (and not so consistent) paradigm of coresets, which are some kind of "compressed representation" of the input set F. Using traditional techniques, a coreset usually implies an LTAS (linear time approximation scheme) for the corresponding optimization problem, which can be computed in parallel, via one pass over the data, and using only polylogarithmic space (i.e, in the streaming model). For several function families F for which coresets are known not to exist, or the corresponding (approximate) optimization problems are hard, our framework yields bicriteria approximations, or coresets that are large, but contained in a low-dimensional space. We demonstrate our unified framework by applying it on projective clustering problems. We obtain new coreset constructions and significantly smaller coresets, over the ones that appeared in the literature during the past years, for problems such as: k-Median [Har-Peled and Mazumdar,STOC'04], [Chen, SODA'06], [Langberg and Schulman, SODA'10]; k-Line median [Feldman, Fiat and Sharir, FOCS'06], [Deshpande and Varadarajan, STOC'07]; Projective clustering [Deshpande et al., SODA'06] [Deshpande and Varadarajan, STOC'07]; Linear l p regression [Clarkson, Woodruff, STOC'09 ]; Low-rank approximation [Sarlos, FOCS'06]; Subspace approximation [Shyamalkumar and Varadarajan, SODA'07], [Feldman, Monemizadeh, Sohler and Woodruff, SODA'10], [Deshpande, Tulsiani, and Vishnoi, SODA'11]. The running times of the corresponding optimization problems are also significantly improved. We show how to generalize the results of our framework for squared distances (as in k-mean), distances to the qth power, and deterministic constructions.

Original languageEnglish
Title of host publicationSTOC'11 - Proceedings of the 43rd ACM Symposium on Theory of Computing
Pages569-578
Number of pages10
DOIs
StatePublished - 2011
Externally publishedYes
Event43rd ACM Symposium on Theory of Computing, STOC'11 - San Jose, CA, United States
Duration: 6 Jun 20118 Jun 2011

Publication series

NameProceedings of the Annual ACM Symposium on Theory of Computing
ISSN (Print)0737-8017

Conference

Conference43rd ACM Symposium on Theory of Computing, STOC'11
Country/TerritoryUnited States
CitySan Jose, CA
Period6/06/118/06/11

Keywords

  • approximating
  • clustering
  • coresets
  • cur
  • epsilon-approximation
  • epsilon-nets
  • k-means
  • k-median
  • pac-learning
  • pca
  • regression
  • svd

ASJC Scopus subject areas

  • Software

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