A Complete Characterization of Tensorizable f-divergences
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 χ2-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.
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BibTeX
@misc{cruz2026complete,
title = {{A Complete Characterization of Tensorizable $f$-divergences}},
author = {Rodrigo Cruz and Flavio P. Calmon and Qian Yu},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28556},
eprint = {2608.28556},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28556},
}
RIS
TY - UNPB
TI - A Complete Characterization of Tensorizable f-divergences
AU - Rodrigo Cruz
AU - Flavio P. Calmon
AU - Qian Yu
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28556
UR - https://arxiv.org/abs/2608.28556
AB - 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.
ER -