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Metric Structures on Datasets: Stability and Classification of Algorithms

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Computer Analysis of Images and Patterns (CAIP 2011)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 6855))

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Abstract

Several methods in data and shape analysis can be regarded as transformations between metric spaces. Examples are hierarchical clustering methods, the higher order constructions of computational persistent topology, and several computational techniques that operate within the context of data/shape matching under invariances.

Metric geometry, and in particular different variants of the Gromov-Hausdorff distance provide a point of view which is applicable in different scenarios. The underlying idea is to regard datasets as metric spaces, or metric measure spaces (a.k.a. mm-spaces, which are metric spaces enriched with probability measures), and then, crucially, at the same time regard the collection of all datasets as a metric space in itself. Variations of this point of view give rise to different taxonomies that include several methods for extracting information from datasets.

Imposing metric structures on the collection of all datasets could be regarded as a ”soft” construction. The classification of algorithms, or the axiomatic characterization of them, could be achieved by imposing the more ”rigid” category structures on the collection of all finite metric spaces and demanding functoriality of the algorithms. In this case, one would hope to single out all the algorithms that satisfy certain natural conditions, which would clarify the landscape of available methods. We describe how using this formalism leads to an axiomatic description of many clustering algorithms, both flat and hierarchical.

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Authors and Affiliations

  1. Department of Mathematics, Stanford University, California, USA

    Facundo Mémoli

  2. Department of Computer Science, The University of Adelaide, Australia

    Facundo Mémoli

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  1. Facundo Mémoli
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Editors and Affiliations

  1. Dpto. Matematica Aplicada I, Escuaela Técnica Superior de Ingeniería Informática, Universite de Sevilla, Avda. Reina Mercedes, s/n, 41012, Sevilla, Spain

    Pedro Real

  2. Departamento de Matemática Aplicada I, Escuela Técnica Superior de Ingeniería Informática, University of Seville, Avenida Reina Mercedes s/n, 41012, Sevilla, Spain

    Daniel Diaz-Pernil  & Helena Molina-Abril  & 

  3. Departamento de Didáctica de la Mathemática y de las CC.Experimentales, Escuela Universitaria de, Universidad del País Vasco-Esukal Herriko Unibertsitatea, Ramón y Cajal, 72, 48014, Bilbao, (Bizcaia), Spain

    Ainhoa Berciano

  4. Institute of Computer Graphics and Algorithms, Pattern Recognition and Image Processing Group, Vienna University of Technology, Favoritenstraße 9/186-3, 1040, Vienna, Austria

    Walter Kropatsch

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Mémoli, F. (2011). Metric Structures on Datasets: Stability and Classification of Algorithms. In: Real, P., Diaz-Pernil, D., Molina-Abril, H., Berciano, A., Kropatsch, W. (eds) Computer Analysis of Images and Patterns. CAIP 2011. Lecture Notes in Computer Science, vol 6855. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23678-5_1

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