Invariant Coordinate Selection and Fisher discriminant subspace beyond the case of two group

Colombe Becquart, Aurore Archimbaud (), Anne Ruiz-Gazen, Luka Prlić and Klaus Nordhausen
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Colombe Becquart: TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - Comue de Toulouse - Communauté d'universités et établissements de Toulouse - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, UT Capitole - Université Toulouse Capitole - Comue de Toulouse - Communauté d'universités et établissements de Toulouse
Aurore Archimbaud: TBS - Toulouse Business School
Anne Ruiz-Gazen: TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - Comue de Toulouse - Communauté d'universités et établissements de Toulouse - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, UT Capitole - Université Toulouse Capitole - Comue de Toulouse - Communauté d'universités et établissements de Toulouse
Luka Prlić: TSE-R - Toulouse School of Economics - UT Capitole - Université Toulouse Capitole - Comue de Toulouse - Communauté d'universités et établissements de Toulouse - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique - INRAE - Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement, UT Capitole - Université Toulouse Capitole - Comue de Toulouse - Communauté d'universités et établissements de Toulouse
Klaus Nordhausen: Department of Mathematics and Statistics [Helsinki] - Falculty of Science [Helsinki] - Helsingin yliopisto = Helsingfors universitet = University of Helsinki

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Abstract: Invariant Coordinate Selection (ICS) is a multivariate technique that relies on the simultaneous diagonalization of two scatter matrices. It serves various purposes, including its use as a dimension reduction tool prior to clustering or outlier detection. ICS's theoretical foundation establishes why and when the identified subspace should contain relevant information by demonstrating its connection with the Fisher discriminant subspace (FDS). These general results have been examined in detail primarily for specific scatter combinations within a two-cluster framework. In this study, we expand these investigations to include more clusters and scatter combinations. Our analysis reveals the importance of distinguishing whether the group centers matrix has full rank. In the full-rank case, we establish deeper connections between ICS and FDS. We provide a detailed study of these relationships for three clusters when the group centers matrix has full rank and when it does not. Based on these expanded theoretical insights and supported by numerical studies, we conclude that ICS is indeed suitable for recovering the FDS under very general settings and cases of failure seem rare.

Keywords: Subspace estimation; Simultaneous diagonalization; Scatter matrix; Mixture of elliptical distributions; Dimension reduction (search for similar items in EconPapers)
Date: 2026-01
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Published in Journal of Multivariate Analysis, 2026, Vol. 211 (n°105521), ⟨10.1016/j.jmva.2025.105521⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-05482136

DOI: 10.1016/j.jmva.2025.105521

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