Paper: Bregman divergences as power distances, enabling Euclidean NN queries
FrnkNlsn · x · 2026-10-19
Frank Nielsen's paper (arXiv:2607.24197, 47 pages) shows Bregman divergences between parameters can be expressed as power distances between weighted points, with several applications:
- Nearest-neighbor queries: Bregman closest/farthest neighbor queries reduce to Euclidean neighbor queries in one extra dimension after weight offsetting, so off-the-shelf Euclidean/ANN libraries work directly; the figure shows d-dim KL proximity queries become d+1-dim Euclidean NN queries.
- Minimum enclosing balls: Bregman MEBs are equivalent to double-sided weight power MEBs and can be approximated via a single-sided relaxation; an exact LP-type algorithm and an arbitrarily accurate Frank-Wolfe scheme are given.
- Voronoi connection: potential-function liftings for Bregman Voronoi diagrams reinterpret as classical paraboloid liftings, linking them to power diagrams.
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