{"title":"最优概率度量分解的瓦瑟斯坦梯度流","authors":"Jiangze Han, Chris Ryan, Xin T. Tong","doi":"10.2139/ssrn.4831118","DOIUrl":null,"url":null,"abstract":"We examine the infinite-dimensional optimization problem of finding a decomposition of a probability measure into K probability sub-measures to minimize specific loss functions inspired by applications in clustering and user grouping. We analytically explore the structures of the support of optimal sub-measures and introduce algorithms based on Wasserstein gradient flow, demonstrating their convergence. Numerical results illustrate the implementability of our algorithms and provide further insights.","PeriodicalId":507782,"journal":{"name":"SSRN Electronic Journal","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Wasserstein gradient flow for optimal probability measure decomposition\",\"authors\":\"Jiangze Han, Chris Ryan, Xin T. Tong\",\"doi\":\"10.2139/ssrn.4831118\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"We examine the infinite-dimensional optimization problem of finding a decomposition of a probability measure into K probability sub-measures to minimize specific loss functions inspired by applications in clustering and user grouping. We analytically explore the structures of the support of optimal sub-measures and introduce algorithms based on Wasserstein gradient flow, demonstrating their convergence. Numerical results illustrate the implementability of our algorithms and provide further insights.\",\"PeriodicalId\":507782,\"journal\":{\"name\":\"SSRN Electronic Journal\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-06-03\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SSRN Electronic Journal\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.2139/ssrn.4831118\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SSRN Electronic Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.2139/ssrn.4831118","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
摘要
受聚类和用户分组应用的启发,我们研究了一个无限维优化问题,即如何将一个概率度量分解为 K 个概率子度量,以最小化特定的损失函数。我们分析探索了最优子度量支持的结构,并引入了基于瓦瑟斯坦梯度流的算法,证明了它们的收敛性。数值结果表明了我们算法的可实施性,并提供了进一步的见解。
Wasserstein gradient flow for optimal probability measure decomposition
We examine the infinite-dimensional optimization problem of finding a decomposition of a probability measure into K probability sub-measures to minimize specific loss functions inspired by applications in clustering and user grouping. We analytically explore the structures of the support of optimal sub-measures and introduce algorithms based on Wasserstein gradient flow, demonstrating their convergence. Numerical results illustrate the implementability of our algorithms and provide further insights.