Fuzzy entropic triple k-means
Abstract
This paper proposes fuzzy entropic triple k-means (FE3KM), an entropy-regularized fuzzy partitioning method for three-way data arrays that simultaneously clusters objects, variables, and occasions. Unlike standard fuzzy clustering, which controls fuzziness through a nonlinear exponent m, FE3KM embeds memberships linearly within a Least-Squares (LS) objective, using row-stochastic membership matrices and entropy regularization to represent uncertainty in cluster assignments. This linear structure is what enables an exact decomposition of total deviance into within- and between-cluster components, further attributable to each mode (objects, variables, occasions) and to individual clusters within each mode. This is an interpretive property unavailable to exponent-m fuzzy models. We derive Alternating Least-Squares algorithms for FE3KM, including an accelerated variant for large arrays, and formally relate the method to tandem clustering procedures and to Tucker3/three-mode partitioning models. A simulation study assesses recovery accuracy, robustness to noise and fuzziness mis-specification, and performance against tandem and three-way clustering baselines. An application to a benchmark three-way TV ratings dataset demonstrates how FE3KM recovers interpretable object-variable-occasion structures and quantifies the contribution of each cluster and mode to overall data variability.
Related
- How to approximate the flat spectral triple of a quantum torus by fuzzy tori : a twisted tale
- Invariant means, coset correct means and the Hahn-Banach Theorem
- Quaternionic Buzano-Dragomir Inequality and Deutsch Entropic Uncertainty Principle
- Mostow-Type Decompositions and Geometric Means for Symmetric Cones
- Quantitative stability of classical, entropic, and doubly entropic Wasserstein barycenters
BibTeX
@misc{schenone2026fuzzy,
title = {{Fuzzy entropic triple k-means}},
author = {Mariaelena Bottazzi Schenone and Roberto Rocci and Maurizio Vichi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21538},
eprint = {2609.21538},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21538},
}
RIS
TY - UNPB
TI - Fuzzy entropic triple k-means
AU - Mariaelena Bottazzi Schenone
AU - Roberto Rocci
AU - Maurizio Vichi
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21538
UR - https://arxiv.org/abs/2609.21538
AB - This paper proposes fuzzy entropic triple k-means (FE3KM), an entropy-regularized fuzzy partitioning method for three-way data arrays that simultaneously clusters objects, variables, and occasions. Unlike standard fuzzy clustering, which controls fuzziness through a nonlinear exponent m, FE3KM embeds memberships linearly within a Least-Squares (LS) objective, using row-stochastic membership matrices and entropy regularization to represent uncertainty in cluster assignments. This linear structure is what enables an exact decomposition of total deviance into within- and between-cluster components, further attributable to each mode (objects, variables, occasions) and to individual clusters within each mode. This is an interpretive property unavailable to exponent-m fuzzy models. We derive Alternating Least-Squares algorithms for FE3KM, including an accelerated variant for large arrays, and formally relate the method to tandem clustering procedures and to Tucker3/three-mode partitioning models. A simulation study assesses recovery accuracy, robustness to noise and fuzziness mis-specification, and performance against tandem and three-way clustering baselines. An application to a benchmark three-way TV ratings dataset demonstrates how FE3KM recovers interpretable object-variable-occasion structures and quantifies the contribution of each cluster and mode to overall data variability.
ER -