Frank Nielsen Defines Curved Bregman Divergences, Linking Cosine Distance to Squared Euclidean
FrnkNlsn · x · 2026-09-18
Frank Nielsen's arXiv paper introduces curved Bregman divergences—Bregman divergences restricted to non-affine parameter subspaces—analogous to curved exponential families in statistics.
Key points:
- Canonical example: cosine dissimilarity between normalized vectors is a curved squared Euclidean divergence
- Proves the barycenter under a curved Bregman divergence equals the right Bregman projection of the full-space barycenter onto the subspace
- Applications include symmetrized and pointwise symmetrized Bregman divergences and KL divergence between circular complex normal distributions
- Shows α-divergences are representational curved Bregman divergences via α-embeddings of the probability simplex
- Presents an efficient calculation method
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