Clustering above Exponential Families with Tempered Exponential Measures

TL;DR

This paper introduces tempered exponential measures (TEM), a new generalization of exponential families, enabling robust clustering beyond traditional models and addressing limitations of existing generalized exponential families.

Contribution

The paper proposes TEM, a novel framework that extends exponential families for clustering, improving robustness and maintaining analytic simplicity.

Findings

TEM enhances robustness of clustering methods.

TEM maintains a simple analytic form for population minimizers.

The framework generalizes existing exponential family models.

Abstract

The link with exponential families has allowed $k$-means clustering to be generalized to a wide variety of data generating distributions in exponential families and clustering distortions among Bregman divergences. Getting the framework to work above exponential families is important to lift roadblocks like the lack of robustness of some population minimizers carved in their axiomatization. Current generalisations of exponential families like $q$-exponential families or even deformed exponential families fail at achieving the goal. In this paper, we provide a new attempt at getting the complete framework, grounded in a new generalisation of exponential families that we introduce, tempered exponential measures (TEM). TEMs keep the maximum entropy axiomatization framework of $q$-exponential families, but instead of normalizing the measure, normalize a dual called a co-distribution. Numerous interesting properties arise for clustering such as improved and controllable robustness for population minimizers, that keep a simple analytic form.