Squared Mahalanobis distance is the unique symmetric Bregman divergence, yielding four dual forms
FrnkNlsn · x · 2026-11-09
Frank Nielsen shares a neat ML-theory observation: the squared Mahalanobis distance is the unique symmetric Bregman divergence.
- It follows from the framework in his arXiv paper "Bregman Voronoi Diagrams" (0709.2196, with Boissonnat and Nock), where Bregman divergences cover squared Euclidean distance and entropy-based divergences, enabling information-theoretic Voronoi diagrams in statistical parametric spaces.
- Using the duality of Bregman divergence parameterizations via the Legendre-Fenchel transform, a Gaussian with F(θ)=½θᵀΣ⁻¹θ yields four dual Mahalanobis distances, corresponding to swaps between natural parameters η=Σ⁻¹θ and mean parameters θ.
- He gives the closed form ΔΣ(μ₁,μ₂)=√((μ₂−μ₁)ᵀΣ⁻¹(μ₂−μ₁)) and references related work including 1808.02771.
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