Generalized Euler Logarithm and its Applications in Machine Learning: Natural Gradient, Backpropagation, Generalized EG, Mirror Descent and OLPS

ArXi:2502.17500v3 Announce Type: replace-cross This paper investigates in depth the fundamental properties of the two-parameter generalized Euler logarithm and its inverse, the associated deformed $(a,b)$-exponential function. We systematically clarify the parameter domains that guarantee monotonicity, concavity, and invertibility, derive series and integral representations, and provide explicit links to a broad class of one- and two-parameter deformations, including Tsallis, Kaniadakis, Schw\"ammle--Tsallis, Kaniadakis--Scarfone, and Tempesta-type logarithms and their inverse exponentials.