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[2504.05654] Curved representational Bregman divergences and their applications
Summary
This article introduces curved representational Bregman divergences, exploring their mathematical foundations and applications in information theory and machine learning.
Why It Matters
Understanding curved Bregman divergences is crucial for advancing statistical methods in machine learning. This work provides new insights into parameter spaces and divergence calculations, potentially enhancing algorithms in data science and AI applications.
Key Takeaways
- Curved Bregman divergences are defined for non-affine parameter subspaces, expanding traditional Bregman divergence concepts.
- The paper presents efficient methods for calculating intersections of $$-divergence spheres, which can improve computational efficiency in statistical models.
- Examples include symmetrized Bregman divergences and their applications in circular complex normal distributions.
Computer Science > Information Theory arXiv:2504.05654 (cs) [Submitted on 8 Apr 2025 (v1), last revised 17 Feb 2026 (this version, v4)] Title:Curved representational Bregman divergences and their applications Authors:Frank Nielsen View a PDF of the paper titled Curved representational Bregman divergences and their applications, by Frank Nielsen View PDF HTML (experimental) Abstract: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-dimensi...
