{"paper":{"title":"Monge-Kantorovich Fitting With Sobolev Budgets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"cs.LG","authors_text":"Forest Kobayashi, Jonathan Hayase, Young-Heon Kim","submitted_at":"2024-09-25T01:30:16Z","abstract_excerpt":"Given $m < n$, we consider the problem of ``best'' approximating an $n\\text{-d}$ probability measure $\\rho$ via an $m\\text{-d}$ measure $\\nu$ such that $\\mathrm{supp}\\ \\nu$ has bounded total ``complexity.'' When $\\rho$ is concentrated near an $m\\text{-d}$ set we may interpret this as a manifold learning problem with noisy data. However, we do not restrict our analysis to this case, as the more general formulation has broader applications.\n  We quantify $\\nu$'s performance in approximating $\\rho$ via the Monge-Kantorovich (also called Wasserstein) $p$-cost $\\mathbb{W}_p^p(\\rho, \\nu)$, and const"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.16541","kind":"arxiv","version":2},"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/2409.16541/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"}