A Complete Characterization of Tensorizable $f$-divergences

2026-08-28

AI Summary

本文解决了f-散度的张量化问题，通过精炼张量化的形式化定义，并证明所有可张量化的f-散度都具有由单一参数表征的多重仿射形式。

Abstract

Csiszar's formulation of the $f$-divergence introduced a vast family of functionals for quantifying dissimilarity between probability distributions. However, many applications in statistics and information theory rely only on a few $f$-divergences, such as the Kullback-Leibler divergence, the $χ^2$-divergence, and the squared Hellinger distance. These divergences are especially useful because they admit simple compositional formulas under product measures, a property sometimes referred to as tensorization. In this work, we refine a formalism of tensorization previously introduced in the literature. Then, we show that any possible tensorization formula has a multi-affine form characterized by a single parameter, and identify all tensorizable $f$-divergences under our adopted notion of tensorization.

Problem

• f-divergences
• tensorization
• probability distributions

Innovation

• tensorization
• f-divergences
• multi-affine form