{"paper":{"title":"Rigidity and Quantitative Stability of the Sliced Wasserstein Deficit","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":["math.FA","math.PR","math.SP"],"primary_cat":"math.MG","authors_text":"Bang-Xian Han","submitted_at":"2026-05-25T06:05:05Z","abstract_excerpt":"The sliced Wasserstein distance $SW_2(\\mu,\\nu)$ compares high-dimensional probability measures by averaging one-dimensional optimal transport distances over linear projections. Although sliced Wasserstein distances are now standard computational tools in statistics, imaging, and machine learning, the rigidity behind the elementary comparison \\[\n  SW_2^2(\\mu,\\nu)\\leq \\frac1d W_2^2(\\mu,\\nu) \\] has not been systematically studied.\n  Let $\\mu,\\nu\\in\\mathcal P_2(\\mathbb R^d)$, $d\\ge2$, with $\\mu\\ll\\mathcal L^d$, and define the sliced Wasserstein deficit by \\[\n  {\\mathrm D}(\\mu,\\nu):=\\frac1d W_2^2(\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2605.25453","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/2605.25453/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"}