{"paper":{"title":"The Blessing and Curse of Curvature in Higher-Order Optimization: Horospherical versus Geodesic Convexity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.OC","authors_text":"Andi Han","submitted_at":"2026-08-07T02:27:02Z","abstract_excerpt":"We study deterministic Riemannian $p$-th-order oracle complexity (for $p\\ge2$) on Hadamard manifolds, under strong horospherical ($h$)-convexity and strong geodesic ($g$)-convexity. The two notions agree in the Euclidean space. On a curved Hadamard manifold, $h$-convexity is a stronger notion than $g$-convexity and supplies global horospherical information.\n  Writing the $p$-th-order condition parameter $Q_p=L_pR^{p-1}/\\mu$, we obtain the Euclidean-optimal rate $Q_p^{2/(3p+1)}$ for strongly $h$-convex objectives on every Hadamard manifold, with a matching fixed-curvature lower bound. On hyperb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.06719","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.06719/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}