# Generalising maximum mean discrepancy: kernelised functional Bregman divergences

arXiv: https://arxiv.org/abs/2604.24047 · HTML: https://mlsanity.com/paper/2604.24047/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 13 of its 46 references that are in the map.
- First posted on arXiv 27 Apr 2026.

## Builds on (most cited first, 13 of 13)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Moment Matching Networks](https://mlsanity.com/paper/1502.02761/) (2015)
- [MMD GAN: Towards Deeper Understanding of Moment Matching Network](https://mlsanity.com/paper/1705.08584/) (2017)
- [Maximum Mean Discrepancy Gradient Flow](https://mlsanity.com/paper/1906.04370/) (2019)
- [Learning with Fenchel-Young Losses](https://mlsanity.com/paper/1901.02324/) (2019)
- [An elementary introduction to information geometry](https://mlsanity.com/paper/1808.08271/) (2018)
- [Relative Loss Bounds for On-line Density Estimation with the Exponential Family of Distributions](https://mlsanity.com/paper/1301.6677/) (2013)
- [Generalized Variational Inference: Three arguments for deriving new Posteriors](https://mlsanity.com/paper/1904.02063/) (2019)
- [The Information Geometry of Mirror Descent](https://mlsanity.com/paper/1310.7780/) (2013)
- [Finite sample properties of parametric MMD estimation: robustness to misspecification and dependence](https://mlsanity.com/paper/1912.05737/) (2019)
- [Generalized Variational Inference in Function Spaces: Gaussian Measures meet Bayesian Deep Learning](https://mlsanity.com/paper/2205.06342/) (2022)
- [Bregman Divergences for Infinite Dimensional Covariance Matrices](https://mlsanity.com/paper/1403.4334/) (2014)
- [Distributed Bayesian Estimation in Sensor Networks: Consensus on Marginal Densities](https://mlsanity.com/paper/2312.01227/) (2023)

## Abstract (arXiv)

Bregman divergences play a pivotal role in statistics, machine learning and computational information geometry. Particularly in the context of machine learning, they are central to clustering, exponential families, parameter estimation and optimisation, among other things. Despite this, the full toolkit of Hilbert spaces and in particular reproducing kernel Hilbert spaces have not been systematically developed and applied to functional Bregman divergences, where points are functions rather than finite-dimensional parameter vectors. While other types of functional Bregman divergences have been studied, these are typically in a Banach space rather than more directly aligned with kernel methods and Hilbert-space geometry commonly used in machine learning. We consider functional Bregman divergences on a Hilbert space, where the self-dual pairing and Riesz representer afford us particularly convenient calculus. Further specialising Bregman generators as a composition involving a kernel mean embedding makes such divergences easy to estimate. We discuss applications in clustering, universal estimation, robust estimation and generative modelling, and contrast our approach with other types of Bregman divergences.