# On H\"older projective divergences

## TL;DR

This paper introduces a new framework for constructing statistical divergences based on inequalities, specifically H"older inequalities, leading to novel, rescaling-invariant distances that outperform traditional divergences in clustering tasks.

## Contribution

The paper develops two new classes of H"older divergences and pseudo-divergences, providing closed-form formulas for exponential family distributions and demonstrating their effectiveness in clustering.

## Key findings

- New H"older divergences encapsulate Cauchy-Schwarz divergence
- Closed-form formulas for exponential family distributions
- H"older divergences outperform Cauchy-Schwarz in clustering

## Abstract

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.