A Structural Characterization of Entropy Functionals

What the math gives to ML

The paper supplies a structural rule for selecting entropy and divergence families rather than choosing Shannon, Rényi, or Tsallis by convention: admissibility is controlled by convexity of the transformed mean generator, specifically convexity of t \mapsto g(1/t) for increasing g and concavity for decreasing g. This criterion is equivalent to strict convexity of an associated Csiszár generator, which gives data processing under arbitrary Markov kernels and interpretable equality conditions. A practical transfer is a learnable family of divergence losses whose generator is constrained to remain convex, allowing task-adaptive regularization while retaining contraction under stochastic augmentations, token merging, pooling, or teacher-student channels.