Original title: The Tempered Hilbert Simplex Distance and Its Application To Non-linear Embeddings of TEMs
Authors: Ehsan Amid, Frank Nielsen, Richard Nock, Manfred K. Warmuth
The article delves into Tempered Exponential Measures (TEMs), a versatile set of distributions maximizing tempered entropy. Calculus on TEMs operates through deformed arithmetic induced by modified logarithms. They introduce three finite discrete TEM parameterizations via Legendre functions linked to the negative tempered entropy. Notably, they establish an isometry between these parameterizations, forming a tempered Hilbert co-simplex distance, akin to Hilbert geometry. This distance mirrors the tempered Funk distance under a $t$-symmetrization scheme. They introduce “t-lengths” for smooth curves in a Finsler manifold, motivated by this construction. The study showcases their generalized structure across diverse contexts and examines its differentiable approximations’ quality for machine learning optimization through numerical evaluations. This work unfolds a novel approach in understanding TEMs and their applications, offering potential implications for various domains, including machine learning optimization.
Original article: https://arxiv.org/abs/2311.13459