# On H"older projective divergences

arXiv: https://arxiv.org/abs/1701.03916 · HTML: https://mlsanity.com/paper/1701.03916/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jan 2017.

## Built on by (most cited first, 5 of 5)

- [Semi-Supervised Disease Classification based on Limited Medical Image Data](https://mlsanity.com/paper/2405.04295/) (2024)
- [Fast approximations of the Jeffreys divergence between univariate Gaussian mixture models via exponential polynomial densities](https://mlsanity.com/paper/2107.05901/) (2021)
- [Robust Divergence Learning for Missing-Modality Segmentation](https://mlsanity.com/paper/2411.08305/) (2024)
- [Uncertainty Quantification via H"older Divergence for Multi-View Representation Learning](https://mlsanity.com/paper/2411.00826/) (2024)
- [Relationship between H"older Divergence and Functional Density Power Divergence: Intersection and Generalization](https://mlsanity.com/paper/2504.17008/) (2025)

## Abstract (arXiv)

We describe a framework to build distances by measuring the tightness of inequalities, and introduce the notion of proper statistical divergences and improper pseudo-divergences. We then consider the H"older ordinary and reverse inequalities, and present two novel classes of H"older divergences and pseudo-divergences that both encapsulate the special case of the Cauchy-Schwarz divergence. We report closed-form formulas for those statistical dissimilarities when considering distributions belonging to the same exponential family provided that the natural parameter space is a cone (e.g., multivariate Gaussians), or affine (e.g., categorical distributions). Those new classes of H"older distances are invariant to rescaling, and thus do not require distributions to be normalized. Finally, we show how to compute statistical H"older centroids with respect to those divergences, and carry out center-based clustering toy experiments on a set of Gaussian distributions that demonstrate empirically that symmetrized H"older divergences outperform the symmetric Cauchy-Schwarz divergence.