{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/manifold-learning-in-wasserstein-space","title":"Manifold learning in Wasserstein space","arxiv_id":"2311.08549","date":"2023-11-14","proceeding":null,"authors":["Keaton Hamm","Caroline Moosmüller","Bernhard Schmitzer","Matthew Thorpe"],"abstract":"This paper aims at building the theoretical foundations for manifold learning algorithms in the space of absolutely continuous probability measures $\\mathcal{P}_{\\mathrm{a.c.}}(\\Omega)$ with $\\Omega$ a compact and convex subset of $\\mathbb{R}^d$, metrized with the Wasserstein-2 distance $\\mathbb{W}$. We begin by introducing a construction of submanifolds $\\Lambda$ in $\\mathcal{P}_{\\mathrm{a.c.}}(\\Omega)$ equipped with metric $\\mathbb{W}_\\Lambda$, the geodesic restriction of $\\mathbb{W}$ to $\\Lambda$. In contrast to other constructions, these submanifolds are not necessarily flat, but still allow for local linearizations in a similar fashion to Riemannian submanifolds of $\\mathbb{R}^d$. We then show how the latent manifold structure of $(\\Lambda,\\mathbb{W}_{\\Lambda})$ can be learned from samples $\\{\\lambda_i\\}_{i=1}^N$ of $\\Lambda$ and pairwise extrinsic Wasserstein distances $\\mathbb{W}$ on $\\mathcal{P}_{\\mathrm{a.c.}}(\\Omega)$ only. In particular, we show that the metric space $(\\Lambda,\\mathbb{W}_{\\Lambda})$ can be asymptotically recovered in the sense of Gromov--Wasserstein from a graph with nodes $\\{\\lambda_i\\}_{i=1}^N$ and edge weights $W(\\lambda_i,\\lambda_j)$. In addition, we demonstrate how the tangent space at a sample $\\lambda$ can be asymptotically recovered via spectral analysis of a suitable ``covariance operator'' using optimal transport maps from $\\lambda$ to sufficiently close and diverse samples $\\{\\lambda_i\\}_{i=1}^N$. The paper closes with some explicit constructions of submanifolds $\\Lambda$ and numerical examples on the recovery of tangent spaces through spectral analysis.","url_abs":"https://arxiv.org/abs/2311.08549v3","url_pdf":"https://arxiv.org/pdf/2311.08549v3.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"manifold-learning-in-wasserstein-space","repo_url":"https://github.com/bernhard-schmitzer/w2manifolds","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"BSD-3-Clause"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2311.08549","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2311.08549"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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