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The main OT tool on these spaces is the Wasserstein distance which suffers from a heavy computational burden. On Euclidean spaces, a popular alternative is the Sliced-Wasserstein distance, which leverages a closed-form solution of the Wasserstein distance in one dimension, but which is not readily available on manifolds. In this work, we derive general constructions of Sliced-Wasserstein distances on Cartan-Hadamard manifolds, Riemannian manifolds with non-positive curvature, which include among others Hyperbolic spaces or the space of Symmetric Positive Definite matrices. Then, we propose different applications. Additionally, we derive non-parametric schemes to minimize these new distances by approximating their Wasserstein gradient flows.","url_abs":"https://arxiv.org/abs/2403.06560v1","url_pdf":"https://arxiv.org/pdf/2403.06560v1.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":"sliced-wasserstein-distances-and-flows-on","repo_url":"https://github.com/clbonet/sliced-wasserstein_distances_and_flows_on_cartan-hadamard_manifolds","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"pytorch","reach":null}],"tasks":[],"methods":[{"method_slug":"non","method_name":"NON"}],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":"https://syntology.ai/paper/2403.06560","atlas_url":"https://app.syntology.ai/?focus=2403.06560","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"2403.06560"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-25T09:33:49+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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