The Tempered Hilbert Simplex Distance and Its Application To Non-linear Embeddings of TEMs
This work provides a novel geometric framework for TEMs, which could impact machine learning optimization, but it appears incremental as it builds on existing Hilbert geometry concepts.
The paper tackles the problem of generalizing the Hilbert simplex distance to Tempered Exponential Measures (TEMs) by introducing a tempered Hilbert co-simplex distance, establishing isometries between parameterizations, and demonstrates its properties and differentiable approximations for optimization in machine learning.
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.