Frank Nielsen paper introduces scale-of-means framework unifying Fréchet and Karcher means
FrnkNlsn · x · 2026-10-19
Frank Nielsen published an arXiv paper (2511.21173) introducing "scales of means" — parametric one-parameter families of means that interpolate between endpoints of an interval.
- Proves any interior point of an open interval on the real line can be interpreted as a Fréchet mean under a suitable metric, extending prior work limited to positive reals via power means
- Demonstrates the framework with scales of exponential means and radical means
- Views the Euclidean line as a 1D Hessian Riemannian manifold, deriving pairs of dual Fréchet/Karcher means linked by convex duality in dual coordinate systems
- Shows squared Hessian metrics in arbitrary dimension reduce to Euclidean geometry whose Riemannian center of mass coincides with multivariate quasi-arithmetic means, i.e., left-sided Bregman centroids
17 pages, spanning computational geometry and information theory, relevant to Bregman divergence theory in ML.
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