{"paper":{"title":"Two-Sample Inference for Gaussian-Smoothed Wasserstein Costs with Finite Moments","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The plug-in estimator for the Gaussian-smoothed Wasserstein cost converges at rates determined by the distributions' polynomial moments.","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Jiaping Yang, Yunxin Zhang","submitted_at":"2026-05-09T17:49:04Z","abstract_excerpt":"Gaussian smoothing has emerged as an effective technique for reducing the sample complexity of optimal transport. In this paper, we study the two-sample plug-in estimator of the Gaussian-smoothed Wasserstein cost \\(T_p^{(\\sigma)}(\\mu,\\nu)=W_p(\\mu*\\gamma_\\sigma,\\nu*\\gamma_\\sigma)^p\\) on \\(\\R^d\\). For fixed smoothing and finite polynomial moments \\(M_{q_\\mu}(\\mu)<\\infty\\), \\(M_{q_\\nu}(\\nu)<\\infty\\), with \\(q_\\mu,q_\\nu>p\\), we establish upper bounds in probability of order \\(\\rho_{q_\\mu,p,d}(m)+\\rho_{q_\\nu,p,d}(n)\\). Here \\(\\rho_{q,p,d}(N)=N^{-(q-p)/(q+d)}\\) for \\(p<q<d+2p\\), \\(N^{-1/2}\\log N\\) a"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"For fixed smoothing and finite polynomial moments M_{q_μ}(μ)<∞, M_{q_ν}(ν)<∞ with q_μ,q_ν>p, we establish upper bounds in probability of order ρ_{q_μ,p,d}(m)+ρ_{q_ν,p,d}(n) where ρ_{q,p,d}(N)=N^{-(q-p)/(q+d)} for p<q<d+2p, N^{-1/2} log N at q=d+2p, and N^{-1/2} for q>d+2p.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The distributions μ and ν possess finite polynomial moments of order q_μ and q_ν strictly greater than p; the smoothing parameter σ is fixed and positive.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Provides probabilistic upper bounds of order depending on moment orders and a central limit theorem for two-sample estimators of Gaussian-smoothed p-Wasserstein distances under finite moments.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The plug-in estimator for the Gaussian-smoothed Wasserstein cost converges at rates determined by the distributions' polynomial moments.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"e34f150c8a309653b4218404d36e2e516813a689794014c43bf8f98d54635abc"},"source":{"id":"2605.09084","kind":"arxiv","version":2},"verdict":{"id":"6c21ad98-5f05-4673-8592-8ccafc7df6c1","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-12T03:20:47.496906Z","strongest_claim":"For fixed smoothing and finite polynomial moments M_{q_μ}(μ)<∞, M_{q_ν}(ν)<∞ with q_μ,q_ν>p, we establish upper bounds in probability of order ρ_{q_μ,p,d}(m)+ρ_{q_ν,p,d}(n) where ρ_{q,p,d}(N)=N^{-(q-p)/(q+d)} for p<q<d+2p, N^{-1/2} log N at q=d+2p, and N^{-1/2} for q>d+2p.","one_line_summary":"Provides probabilistic upper bounds of order depending on moment orders and a central limit theorem for two-sample estimators of Gaussian-smoothed p-Wasserstein distances under finite moments.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The distributions μ and ν possess finite polynomial moments of order q_μ and q_ν strictly greater than p; the smoothing parameter σ is fixed and positive.","pith_extraction_headline":"The plug-in estimator for the Gaussian-smoothed Wasserstein cost converges at rates determined by the distributions' polynomial moments."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.09084/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"claim_evidence","ran_at":"2026-05-20T08:22:01.685924Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T20:38:07.338046Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T13:31:18.992307Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T10:32:02.335300Z","status":"completed","version":"1.0.0","findings_count":0}],"snapshot_sha256":"4d44c4d7a4411d509db7c3228d5f26fccb68fd13a3b36ad2e920696c587a41ef"},"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"}