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The link with exponential families has allowed $k$-means clustering to begeneralized to a wide variety of data generating distributions in exponentialfamilies and clustering distortions among Bregman divergences. Getting theframework to work above exponential families is important to lift roadblockslike the lack of robustness of some population minimizers carved in theiraxiomatization. Current generalisations of exponential families like$q$-exponential families or even deformed exponential families fail atachieving the goal. In this paper, we provide a new attempt at getting thecomplete framework, grounded in a new generalisation of exponential familiesthat we introduce, tempered exponential measures (TEM). TEMs keep the maximumentropy axiomatization framework of $q$-exponential families, but instead ofnormalizing the measure, normalize a dual called a co-distribution. Numerousinteresting properties arise for clustering such as improved and controllablerobustness for population minimizers, that keep a simple analytic form.

Clustering above Exponential Families with Tempered Exponential Measures