{"id":10286,"date":"2026-02-06T07:02:55","date_gmt":"2026-02-06T07:02:55","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2026\/02\/06\/2602-05174\/"},"modified":"2026-02-06T07:02:55","modified_gmt":"2026-02-06T07:02:55","slug":"2602-05174","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2026\/02\/06\/2602-05174\/","title":{"rendered":"Total Variation Rates for Riemannian Flow Matching"},"content":{"rendered":"<p>    Total Variation Rates for Riemannian Flow Matching<br \/>\n \t<BR><br \/>\n<BR><\/BR><br \/>\n    <!-- no image --><br \/>\n \t<BR><br \/>\n<BR><\/BR><\/p>\n<div>arXiv:2602.05174v1 Announce Type: new<br \/>\nAbstract: Riemannian flow matching (RFM) extends flow-based generative modeling to data supported on manifolds by learning a time-dependent tangent vector field whose flow-ODE transports a simple base distribution to the data law. We develop a nonasymptotic Total Variation (TV) convergence analysis for RFM samplers that use a learned vector field together with Euler discretization on manifolds. Our key technical ingredient is a differential inequality governing the evolution of TV between two manifold ODE flows, which expresses the time-derivative of TV through the divergence of the vector-field mismatch and the score of the reference flow; controlling these terms requires establishing new bounds that explicitly account for parallel transport and curvature. Under smoothness assumptions on the population flow-matching field and either uniform (compact manifolds) or mean-square (Hadamard manifolds) approximation guarantees for the learned field, we obtain explicit bounds of the form $mathrm{TV}le C_{mathrm{Lip}},h + C_{varepsilon},varepsilon$ (with an additional higher-order $varepsilon^2$ term on compact manifolds), cleanly separating numerical discretization and learning errors. Here, $h$ is the step-size and $varepsilon$ is the target accuracy. Instantiations yield emph{explicit} polynomial iteration complexities on the hypersphere $S^d$, and on the SPD$(n)$ manifolds under mild moment conditions.<\/div>\n<p> \t<BR><br \/>\n <BR><\/BR><br \/>\n    Yunrui Guan, Krishnakumar Balasubramanian, Shiqian Ma<br \/>\n \t<BR><br \/>\n<BR><\/BR><br \/>\n<a href=\"https:\/\/arxiv.org\/abs\/2602.05174\">Go to original source<\/a><br \/>\n \t<BR><br \/>\n <BR><\/BR><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Total Variation Rates for Riemannian Flow Matching arXiv:2602.05174v1 Announce Type: new Abstract: Riemannian flow matching (RFM) extends flow-based generative modeling to data supported on manifolds by learning a time-dependent tangent vector field whose flow-ODE transports a simple base distribution to the data law. We develop a nonasymptotic Total Variation (TV) convergence analysis for RFM samplers [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,187,113,190,112,191],"tags":[1049,1612,4741],"class_list":["post-10286","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-ai","category-cs-lg","category-math-st","category-stat-ml","category-stat-th","tag-field","tag-flow","tag-manifolds"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/10286"}],"collection":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/comments?post=10286"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/10286\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=10286"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=10286"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=10286"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}