# The Tempered Hilbert Simplex Distance and Its Application To Non-linear Embeddings of TEMs

arXiv: https://arxiv.org/abs/2311.13459 · HTML: https://mlsanity.com/paper/2311.13459/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 35 references that are in the map.
- First posted on arXiv 22 Nov 2023.

## Builds on (most cited first, 5 of 5)

- [Boosting with Tempered Exponential Measures](https://mlsanity.com/paper/2306.05487/) (2023)
- [Non-linear Embeddings in Hilbert Simplex Geometry](https://mlsanity.com/paper/2203.11434/) (2022)
- [Clustering above Exponential Families with Tempered Exponential Measures](https://mlsanity.com/paper/2211.02765/) (2022)
- [Voronoi Diagrams in the Hilbert Metric](https://mlsanity.com/paper/2112.03056/) (2021)
- [Optimal Transport with Tempered Exponential Measures](https://mlsanity.com/paper/2309.04015/) (2023)

## Built on by (most cited first, 1 of 1)

- [Tempered Calculus for ML: Application to Hyperbolic Model Embedding](https://mlsanity.com/paper/2402.04163/) (2024)

## Abstract (arXiv)

Tempered Exponential Measures (TEMs) are a parametric generalization of the exponential family of distributions maximizing the tempered entropy function among positive measures subject to a probability normalization of their power densities. Calculus on TEMs relies on a deformed algebra of arithmetic operators induced by the deformed logarithms used to define the tempered entropy. In this work, we introduce three different parameterizations of finite discrete TEMs via Legendre functions of the negative tempered entropy function. In particular, we establish an isometry between such parameterizations in terms of a generalization of the Hilbert log cross-ratio simplex distance to a tempered Hilbert co-simplex distance. Similar to the Hilbert geometry, the tempered Hilbert distance is characterized as a t-symmetrization of the oriented tempered Funk distance. We motivate our construction by introducing the notion of t-lengths of smooth curves in a tautological Finsler manifold. We then demonstrate the properties of our generalized structure in different settings and numerically examine the quality of its differentiable approximations for optimization in machine learning settings.