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Independence versus indetermination: basis of two canonical clustering criteria

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Abstract

This paper aims at comparing two coupling approaches as basic layers for building clustering criteria, suited for modularizing and clustering very large networks. We briefly use “optimal transport theory” as a starting point, and a way as well, to derive two canonical couplings: “statistical independence” and “logical indetermination”. A symmetric list of properties is provided and notably the so called “Monge’s properties”, applied to contingency matrices, and justifying the \(\otimes \) versus \(\oplus \) notation. A study is proposed, highlighting “logical indetermination”, because it is, by far, lesser known. Eventually we estimate the average difference between both couplings as the key explanation of their usually close results in network clustering.

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Notes

  1. In 1961 Alan Hoffman (IBM Fellow and US Science Academy member) rediscovered Monges’s observation see Hoffman (1963). Hoffman showed that the Hitchcock-Kantorovich transportation problem can be solved by a very simple approach if its underlying cost matrix satisfies those Monge’s properties.

  2. Factually this is the method of S. Lloyd(1957) rewritten by E.W. Forgy (1965) which corresponds to the oldest version of the K-means really used.

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Acknowledgements

We thank the editor and two anonymous referees for their valuable comments.

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Authors and Affiliations

  1. Laboratoire de Probabilités, Statistique et Modélisation | CNRS UMR 8001, Sorbonne Université, Paris, France

    Pierre Bertrand & Michel Broniatowski

  2. Institut de statistique, Sorbonne Université, Paris, France

    Jean-François Marcotorchino

Authors
  1. Pierre Bertrand
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  2. Michel Broniatowski
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  3. Jean-François Marcotorchino
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Correspondence to Pierre Bertrand.

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Bertrand, P., Broniatowski, M. & Marcotorchino, JF. Independence versus indetermination: basis of two canonical clustering criteria. Adv Data Anal Classif 16, 1069–1093 (2022). https://doi.org/10.1007/s11634-021-00484-1

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