Independence versus Indetermination: basis of two canonical clustering criteria

Pierre Jean-Claude Robert Bertrand , Michel Broniatowski (1) , Jean-François Marcotorchino (2)

1 LPSM (UMR_8001) - Laboratoire de Probabilités, Statistique et Modélisation
2 ISUP - Institut de Statistique de l'Université Pierre et Marie Curie

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.

Mots clés: Mathematical Relational Analysis; Logical Indetermination; Coupling Functions; Correlation Clustering; Optimal Transport; Graph Theoretical Approaches

Domaines: Statistiques [math.ST]; Mathématique discrète [cs.DM]; Réseaux sociaux et d'information [cs.SI]; Probabilités [math.PR]

Pierre Jean-Claude Robert Bertrand, Michel Broniatowski, Jean-François Marcotorchino. Independence versus Indetermination: basis of two canonical clustering criteria. Advances in Data Analysis and Classification, In press, 16 (4), ⟨10.1007/s11634-021-00484-1⟩. ⟨hal-02901167v3⟩