{"article_processing_charge":"No","oa":1,"title":"Heat flow, log-concavity, and Lipschitz transport maps","date_updated":"2024-07-31T08:56:38Z","main_file_link":[{"url":"https://doi.org/10.48550/arXiv.2404.15205","open_access":"1"}],"related_material":{"record":[{"status":"for_moderation","id":"17336","relation":"dissertation_contains"}]},"department":[{"_id":"JaMa"}],"status":"public","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","type":"preprint","publication_status":"submitted","year":"2024","abstract":[{"lang":"eng","text":"In this paper we derive estimates for the Hessian of the logarithm\r\n(log-Hessian) for solutions to the heat equation. For initial data in the form\r\nof log-Lipschitz perturbation of strongly log-concave measures, the log-Hessian\r\nadmits an explicit, uniform (in space) lower bound. This yields a new estimate\r\nfor the Lipschitz constant of a transport map pushing forward the standard\r\nGaussian to a measure in this class. Further connections are discussed with\r\nscore-based diffusion models and improved Gaussian logarithmic Sobolev\r\ninequalities. Finally, we show that assuming only fast decay of the tails of\r\nthe initial datum does not suffice to guarantee uniform log-Hessian upper\r\nbounds."}],"date_created":"2024-07-31T08:17:14Z","oa_version":"Preprint","publication":"arXiv","external_id":{"arxiv":["2404.15205"]},"day":"08","doi":"10.48550/arXiv.2404.15205","author":[{"first_name":"Giovanni","full_name":"Brigati, Giovanni","id":"63ff57e8-1fbb-11ee-88f2-f558ffc59cf1","last_name":"Brigati"},{"first_name":"Francesco","full_name":"Pedrotti, Francesco","id":"d3ac8ac6-dc8d-11ea-abe3-e2a9628c4c3c","last_name":"Pedrotti"}],"_id":"17353","corr_author":"1","date_published":"2024-05-08T00:00:00Z","month":"05","citation":{"ieee":"G. Brigati and F. Pedrotti, “Heat flow, log-concavity, and Lipschitz transport maps,” <i>arXiv</i>. .","apa":"Brigati, G., &#38; Pedrotti, F. (n.d.). Heat flow, log-concavity, and Lipschitz transport maps. <i>arXiv</i>. <a href=\"https://doi.org/10.48550/arXiv.2404.15205\">https://doi.org/10.48550/arXiv.2404.15205</a>","ama":"Brigati G, Pedrotti F. Heat flow, log-concavity, and Lipschitz transport maps. <i>arXiv</i>. doi:<a href=\"https://doi.org/10.48550/arXiv.2404.15205\">10.48550/arXiv.2404.15205</a>","chicago":"Brigati, Giovanni, and Francesco Pedrotti. “Heat Flow, Log-Concavity, and Lipschitz Transport Maps.” <i>ArXiv</i>, n.d. <a href=\"https://doi.org/10.48550/arXiv.2404.15205\">https://doi.org/10.48550/arXiv.2404.15205</a>.","ista":"Brigati G, Pedrotti F. Heat flow, log-concavity, and Lipschitz transport maps. arXiv, <a href=\"https://doi.org/10.48550/arXiv.2404.15205\">10.48550/arXiv.2404.15205</a>.","mla":"Brigati, Giovanni, and Francesco Pedrotti. “Heat Flow, Log-Concavity, and Lipschitz Transport Maps.” <i>ArXiv</i>, doi:<a href=\"https://doi.org/10.48550/arXiv.2404.15205\">10.48550/arXiv.2404.15205</a>.","short":"G. Brigati, F. Pedrotti, ArXiv (n.d.)."},"language":[{"iso":"eng"}]}