Entropy-Generated Attention Beyond Softmax and Entmax: Kaniadakis and Reciprocal-Symmetric Abe Operators

Abstract

We derive two attention operators from generalized statistical entropies. Kaniadakis entropy yields an exact full-support normalization whose weights and low-score sensitivities decay algebraically, rather than exponentially as in Softmax or by exact truncation as in entmax. Classical Abe entropy yields an implicit reciprocal-symmetric operator. With $q=e^{\epsilon }$, the involution $q\leftrightarrow q^{-1}$ removes every odd correction about Softmax; we obtain the normalized second- and fourth-order terms, including the deformation of the normalization multiplier. These stationary laws follow from a Fisher-metric Lagrangian on the probability simplex, whose Shannon sector recovers scaled dot-product Softmax. We also give a tangent-gradient test for deciding whether changing the entropy changes the attention profile or only its scale. Rényi and two-parameter Sharma--Mittal entropies retain the Tsallis--entmax inverse-gradient shape, but their global moments make the effective temperature input dependent when the external temperature is fixed. Distinguishing profile-shape equivalence from fixed-parameter operator equivalence separates new normalization shapes from adaptive rescalings and organizes the operators by support, tail behavior, and realization complexity.

10 pages and 1 figure. Substantially revised and expanded version. The previous numerical study has been removed, and the manuscript now develops a generalized entropy-to-attention framework,including Kaniadakis and reciprocal-symmetric Abe operators, Rényi and Sharma-Mittal projections, and a tangent-gradient criterion for operator equivalence. Title changed