The Tempered Hilbert Simplex Distance and Its Application To Non-linear
  Embeddings of TEMs
Ehsan Amid Frank Nielsen Richard Nock Manfred K. Warmuth
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.