{"id":3098,"date":"2025-04-15T07:03:04","date_gmt":"2025-04-15T07:03:04","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2025\/04\/15\/2504-09279\/"},"modified":"2025-04-15T07:03:04","modified_gmt":"2025-04-15T07:03:04","slug":"2504-09279","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2025\/04\/15\/2504-09279\/","title":{"rendered":"No-Regret Generative Modeling via Parabolic Monge-Amp`ere PDE"},"content":{"rendered":"<p>    No-Regret Generative Modeling via Parabolic Monge-Amp`ere PDE<br \/>\n \t<BR><br \/>\n<BR><\/BR><br \/>\n    <!-- no image --><br \/>\n \t<BR><br \/>\n<BR><\/BR><\/p>\n<div>arXiv:2504.09279v1 Announce Type: new<br \/>\nAbstract: We introduce a novel generative modeling framework based on a discretized parabolic Monge-Amp`ere PDE, which emerges as a continuous limit of the Sinkhorn algorithm commonly used in optimal transport. Our method performs iterative refinement in the space of Brenier maps using a mirror gradient descent step. We establish theoretical guarantees for generative modeling through the lens of no-regret analysis, demonstrating that the iterates converge to the optimal Brenier map under a variety of step-size schedules. As a technical contribution, we derive a new Evolution Variational Inequality tailored to the parabolic Monge-Amp`ere PDE, connecting geometry, transportation cost, and regret. Our framework accommodates non-log-concave target distributions, constructs an optimal sampling process via the Brenier map, and integrates favorable learning techniques from generative adversarial networks and score-based diffusion models. As direct applications, we illustrate how our theory paves new pathways for generative modeling and variational inference.<\/div>\n<p> \t<BR><br \/>\n <BR><\/BR><br \/>\n    Nabarun Deb, Tengyuan Liang<br \/>\n \t<BR><br \/>\n<BR><\/BR><br \/>\n<a href=\"https:\/\/arxiv.org\/abs\/2504.09279\">Go to original source<\/a><br \/>\n \t<BR><br \/>\n <BR><\/BR><\/p>\n","protected":false},"excerpt":{"rendered":"<p>No-Regret Generative Modeling via Parabolic Monge-Amp`ere PDE arXiv:2504.09279v1 Announce Type: new Abstract: We introduce a novel generative modeling framework based on a discretized parabolic Monge-Amp`ere PDE, which emerges as a continuous limit of the Sinkhorn algorithm commonly used in optimal transport. Our method performs iterative refinement in the space of Brenier maps using a mirror [&hellip;]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,113,376,190,112,191],"tags":[252,881,660],"class_list":["post-3098","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-lg","category-math-oc","category-math-st","category-stat-ml","category-stat-th","tag-generative","tag-modeling","tag-regret"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/3098"}],"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=3098"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/3098\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=3098"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=3098"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=3098"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}