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

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Ehsan Amid , Frank Nielsen , Richard Nock , Manfred K. Warmuth ·

YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2311.13459

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.

Related Topics

Mathematics

Simplex

Combinatorics

Mathematical Analysis

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Mathematics Simplex Exponential function Pure mathematics Combinatorics Mathematical analysis

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2311.13459
PDF: https://arxiv.org/pdf/2311.13459
OA Status: green
Related Works: 10
OpenAlex ID: https://openalex.org/W4388963693

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DOI: https://doi.org/10.48550/arxiv.2311.13459

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

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Language: en

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Publication year: 2023

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Publication date: 2023-11-22

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Authors: Ehsan Amid, Frank Nielsen, Richard Nock, Manfred K. Warmuth

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Landing page: https://arxiv.org/abs/2311.13459

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Concepts: Mathematics, Simplex, Exponential function, Pure mathematics, Combinatorics, Mathematical analysis

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