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Tempered Exponential Measures (TEMs) are a parametric generalization of theexponential family of distributions maximizing the tempered entropy functionamong positive measures subject to a probability normalization of their powerdensities. Calculus on TEMs relies on a deformed algebra of arithmeticoperators induced by the deformed logarithms used to define the temperedentropy. In this work, we introduce three different parameterizations of finitediscrete TEMs via Legendre functions of the negative tempered entropy function.In particular, we establish an isometry between such parameterizations in termsof a generalization of the Hilbert log cross-ratio simplex distance to atempered Hilbert co-simplex distance. Similar to the Hilbert geometry, thetempered Hilbert distance is characterized as a $t$-symmetrization of theoriented tempered Funk distance. We motivate our construction by introducingthe notion of $t$-lengths of smooth curves in a tautological Finsler manifold.We then demonstrate the properties of our generalized structure in differentsettings and numerically examine the quality of its differentiableapproximations for optimization in machine learning settings.

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