Pierre Bertrand

Chapitre de livre

Some smooth divergences for $\ell_{1}-$approximations

Pierre Bertrand, Wolfgang Stummer, Lecture Notes in Computer Science, Vol. 16033, pp. 359-368, 10/2025

Abstract For some smooth special case of generalized $\varphi-$divergences as well as of new divergences (called scaled shift divergences), we derive approximations of the omnipresent (weighted) $\ell_{1}-$distance and (weighted) $\ell_{1}-$norm.

Keywords Divergence Kullback-Leibler, Divergence analysis, $\ell1-$distance/norm, Generalized $\varphi-$divergences

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Article de revue

Independence versus Indetermination: basis of two canonical clustering criteria

Pierre Bertrand, Michel Broniatowski, Jean-François Marcotorchino, Advances in Data Analysis and Classification, Vol. 16, No. 4, 01/2022

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.

Keywords Graph Theoretical Approaches, Optimal Transport, Correlation Clustering, Coupling Functions, Logical Indetermination, Mathematical Relational Analysis

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