An Anatomical Review of the Power Distance Representation of Bregman Divergences: Comparative Geometry, Emergent Epistemics, and Ontological Implications

This anatomical review dissects Frank Nielsen's 2026 representation of Bregman divergences as equivalent power distances (arXiv:2607.24197). The analysis examines the mixed-parameter lifting that converts Bregman proximity, Voronoi, and minimum-enclosing-ball problems into Laguerre geometry operations solvable by Euclidean nearest-neighbour queries in one additional dimension. The representation is compared and contrasted with twelve leading theoretical frameworks: classical Bregman theory, Amari's dually flat information geometry, Bregman Voronoi diagrams, Aurenhammer's power diagrams, optimal transport (including Bregman–Wasserstein), f- and α-divergences, comparative convexity, exponential-family geometry, Hessian Riemannian manifolds, semi-discrete optimal transport, Fenchel–Young duality, and total/conformal Bregman divergences. From this comparative anatomy an emergent epistemic is identified: the computational reduction of information-geometric queries to mature Euclidean and Laguerre toolchains. The ontological influence is the elevation of weighted points and sectional power diagrams to co-equal status with the original convex potential, thereby reframing the dually flat manifold as a section of a higher-dimensional Euclidean construction. Implications for algorithmic information geometry, machine learning, and geometric computing are discussed.