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Common Geodesics Do Not Guarantee Fisher Consistency of the Structured SVM: Minimal Counterexamples and a Tree-Metric Classification

arXiv · AI, language, vision and robotics · article · Aug 27, 2026 · UTC

A known necessary condition for Fisher consistency of the structured support vector machine requires the task loss to be a metric for which every output triple has a common geodesic point. We show that this condition is not sufficient for the canonical coordinate-wise argmax decoder. A four-output unit star admits an exactly optimal score vector whose maximizers are all strictly non-Bayes, and four outputs are minimal among metrics satisfying the condition. We then completely classify positively weighted tree metrics whose vertex set is the output space: argmax consistency holds if and only if

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