# Curved representational Bregman divergences and their applications

arXiv: https://arxiv.org/abs/2504.05654 · HTML: https://mlsanity.com/paper/2504.05654/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 56 references that are in the map.
- First posted on arXiv 8 Apr 2025.

## Builds on (most cited first, 6 of 6)

- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [On a generalization of the Jensen-Shannon divergence and the JS-symmetrization of distances relying on abstract means](https://mlsanity.com/paper/1904.04017/) (2019)
- [Relative Loss Bounds for On-line Density Estimation with the Exponential Family of Distributions](https://mlsanity.com/paper/1301.6677/) (2013)
- [Bias-variance decompositions: the exclusive privilege of Bregman divergences](https://mlsanity.com/paper/2501.18581/) (2025)
- [Topological Data Analysis with Bregman Divergences](https://mlsanity.com/paper/1607.06274/) (2016)
- [Fast proxy centers for Jeffreys centroids: The Jeffreys-Fisher-Rao and the inductive Gauss-Bregman centers](https://mlsanity.com/paper/2410.14326/) (2024)

## Abstract (arXiv)

By analogy to the terminology of curved exponential families in statistics, we define curved Bregman divergences as Bregman divergences restricted to non-affine parameter subspaces and sub-dimensional Bregman divergences when the restrictions are affine. A common example of curved Bregman divergence is the cosine dissimilarity between normalized vectors: a curved squared Euclidean divergence. We prove that the barycenter of a finite weighted set of parameters under a curved Bregman divergence amounts to the right Bregman projection onto the non-affine subspace of the barycenter with respect to the full Bregman divergence, and interpret a generalization of the weighted Bregman centroid of n parameters as a n-fold sub-dimensional Bregman divergence. We demonstrate the significance of curved Bregman divergences with several examples: (1) symmetrized Bregman divergences, (2) pointwise symmetrized Bregman divergences, and (3) the Kullback-Leibler divergence between circular complex normal distributions. We explain how to reparameterize sub-dimensional Bregman divergences on simplicial sub-dimensional domains. We then consider monotonic embeddings to define representational curved Bregman divergences and show that the α-divergences are representational curved Bregman divergences with respect to α-embeddings of the probability simplex into the positive measure cone. As an application, we report an efficient method to calculate the intersection of a finite set of α-divergence spheres. As an application, we report an efficient method to calculate the intersection of a finite set of α-divergence spheres.