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This paper critically examines the relationship between common geodesics and Fisher consistency in structured support vector machines (SVMs), revealing that the known necessary condition is not sufficient for the argmax decoder. Through the analysis of minimal counterexamples, it is shown that while argmax consistency holds for tree metrics that are paths, branching trees fail to maintain this property under certain distributions. The findings highlight that five outputs are necessary for a full-support counterexample among metrics satisfying the common-geodesic condition, exposing a significant gap in decoder performance within the polyhedral framework.
Common geodesics fail to ensure Fisher consistency, revealing that even optimal score vectors can yield strictly non-Bayes maximizers.
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 the tree is a path. The failure on branching trees is confined to boundary distributions; every tree retains the argmax property at every full-support distribution. Among metrics satisfying the common-geodesic condition, five outputs are necessary and sufficient for a full-support counterexample; $K_{2,3}$ is the smallest member of an infinite $K_{m,n}$ family. We additionally give a full-support counterexample for the three-dimensional Hamming cube. All optimality claims have exact primal-dual certificates. The counterexamples expose a concrete decoder gap: in this polyhedral setting, an embedding can guarantee the existence of a calibrated link without validating a prescribed argmax link on every surrogate-risk minimizer.
