{"id":2414,"date":"2025-03-14T07:02:30","date_gmt":"2025-03-14T07:02:30","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2025\/03\/14\/2503-10576\/"},"modified":"2025-03-14T07:02:30","modified_gmt":"2025-03-14T07:02:30","slug":"2503-10576","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2025\/03\/14\/2503-10576\/","title":{"rendered":"Sample and Map from a Single Convex Potential: Generation using Conjugate Moment Measures"},"content":{"rendered":"

Sample and Map from a Single Convex Potential: Generation using Conjugate Moment Measures
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arXiv:2503.10576v1 Announce Type: new
\nAbstract: A common approach to generative modeling is to split model-fitting into two blocks: define first how to sample noise (e.g. Gaussian) and choose next what to do with it (e.g. using a single map or flows). We explore in this work an alternative route that ties sampling and mapping. We find inspiration in moment measures, a result that states that for any measure $rho$ supported on a compact convex set of $mathbb{R}^d$, there exists a unique convex potential $u$ such that $rho=nabla u,sharp,e^{-u}$. While this does seem to tie effectively sampling (from log-concave distribution $e^{-u}$) and action (pushing particles through $nabla u$), we observe on simple examples (e.g., Gaussians or 1D distributions) that this choice is ill-suited for practical tasks. We study an alternative factorization, where $rho$ is factorized as $nabla w^*,sharp,e^{-w}$, where $w^*$ is the convex conjugate of $w$. We call this approach conjugate moment measures, and show far more intuitive results on these examples. Because $nabla w^*$ is the Monge map between the log-concave distribution $e^{-w}$ and $rho$, we rely on optimal transport solvers to propose an algorithm to recover $w$ from samples of $rho$, and parameterize $w$ as an input-convex neural network.<\/div>\n

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\n Nina Vesseron, Louis B’ethune, Marco Cuturi
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Sample and Map from a Single Convex Potential: Generation using Conjugate Moment Measures arXiv:2503.10576v1 Announce Type: new Abstract: A common approach to generative modeling is to split model-fitting into two blocks: define first how to sample noise (e.g. Gaussian) and choose next what to do with it (e.g. using a single map or flows). We […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,113,112],"tags":[1882,2037,2036],"class_list":["post-2414","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-lg","category-stat-ml","tag-convex","tag-map","tag-rho"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/2414"}],"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=2414"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/2414\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=2414"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=2414"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=2414"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}