# Clustering above Exponential Families with Tempered Exponential Measures

arXiv: https://arxiv.org/abs/2211.02765 · HTML: https://mlsanity.com/paper/2211.02765/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 4 of its 39 references that are in the map.
- First posted on arXiv 4 Nov 2022.

## Builds on (most cited first, 4 of 4)

- [Robust Bi-Tempered Logistic Loss Based on Bregman Divergences](https://mlsanity.com/paper/1906.03361/) (2019)
- [Uniform Concentration Bounds toward a Unified Framework for Robust Clustering](https://mlsanity.com/paper/2110.14148/) (2021)
- [f-GANs in an Information Geometric Nutshell](https://mlsanity.com/paper/1707.04385/) (2017)
- [Bregman Power k-Means for Clustering Exponential Family Data](https://mlsanity.com/paper/2206.10860/) (2022)

## Built on by (most cited first, 6 of 6)

- [Boosting with Tempered Exponential Measures](https://mlsanity.com/paper/2306.05487/) (2023)
- [Optimal Transport with Tempered Exponential Measures](https://mlsanity.com/paper/2309.04015/) (2023)
- [Information Geometry for the Working Information Theorist](https://mlsanity.com/paper/2310.03884/) (2023)
- [The Tempered Hilbert Simplex Distance and Its Application To Non-linear Embeddings of TEMs](https://mlsanity.com/paper/2311.13459/) (2023)
- [Perspectives on Tsallis Statistics for Artificial Intelligence](https://mlsanity.com/paper/2608.01223/) (2026)
- [Tempered Calculus for ML: Application to Hyperbolic Model Embedding](https://mlsanity.com/paper/2402.04163/) (2024)

## Abstract (arXiv)

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.