NDLI: Classifying Clustering Schemes

Content Provider	Springer Nature Link
Author	Carlsson, Gunnar Mémoli, Facundo
Copyright Year	2013
Abstract	Many clustering schemes are defined by optimizing an objective function defined on the partitions of the underlying set of a finite metric space. In this paper, we construct a framework for studying what happens when we instead impose various structural conditions on the clustering schemes, under the general heading of functoriality.Functoriality refers to the idea that one should be able to compare the results of clustering algorithms as one varies the dataset, for example by adding points or by applying functions to it. We show that, within this framework, one can prove a theorem analogous to one of Kleinberg (Becker et al. (eds.), NIPS, pp. 446–453, MIT Press, Cambridge, 2002), in which, for example, one obtains an existence and uniqueness theorem instead of a nonexistence result.We obtain a full classification of all clustering schemes satisfying a condition we refer to as excisiveness. The classification can be changed by varying the notion of maps of finite metric spaces. The conditions occur naturally when one considers clustering as the statistical version of the geometric notion of connected components. By varying the degree of functoriality that one requires from the schemes, it is possible to construct richer families of clustering schemes that exhibit sensitivity to density.
Ending Page	252
Page Count	32
Starting Page	221
File Format	PDF
ISSN	16153375
e-ISSN	16153383
Journal	Foundations of Computational Mathematics
Issue Number	2
Volume Number	13
Language	English
Publisher	Springer-Verlag
Publisher Date	2013-01-09
Publisher Place	New York
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Data analysis Single linkage Economics general Linear and Multilinear Algebras, Matrix Theory Clustering Chaining effect Pattern recognition, speech recognition Classification and discrimination; cluster analysis Numerical Analysis Computer Science Applications of Mathematics Math Applications in Computer Science Functoriality
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Analysis Computational Theory and Mathematics Computational Mathematics

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