这篇论文揭示了结构化SVM中Fisher一致性的微妙边界，给出了最小反例和精确分类。
该论文证明了结构化支持向量机中Fisher一致性的已知必要条件并非充分条件。研究发现，对于四输出单元星形结构，存在严格非贝叶斯的最优得分向量。对于加权树度量，argmax一致性当且仅当树为路径时成立。在满足公共测地线条件的度量中，五个输出足以构造全支撑反例，$K_{2,3}$是无限$K_{m,n}$家族中最小的成员。
Common Geodesics Do Not Guarantee Fisher Consistency of the Structured SVM: Minimal Counterexamples and a Tree-Metric Classification
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.