{"title":"实线上与Gromov-Wasserstein距离有关的分配问题","authors":"Robert Beinert, Cosmas Heiß, G. Steidl","doi":"10.48550/arXiv.2205.09006","DOIUrl":null,"url":null,"abstract":"Let $x_1<\\dots<x_n$ and $y_1<\\dots<y_n$, $n \\in \\mathbb N$, be real numbers. We show by an example that the assignment problem $$ \\max_{\\sigma \\in S_n} F_\\sigma(x,y) := \\frac12 \\sum_{i,k=1}^n |x_i - x_k|^\\alpha \\, |y_{\\sigma(i)} - y_{\\sigma(k)}|^\\alpha, \\quad \\alpha>0, $$ is in general neither solved by the identical permutation (id) nor the anti-identical permutation (a-id) if $n>2 +2^\\alpha$. Indeed the above maximum can be, depending on the number of points, arbitrary far away from $F_\\text{id}(x,y)$ and $F_\\text{a-id}(x,y)$. The motivation to deal with such assignment problems came from their relation to Gromov-Wasserstein divergences which have recently attained a lot of attention.","PeriodicalId":185319,"journal":{"name":"SIAM J. Imaging Sci.","volume":"24 1","pages":"0"},"PeriodicalIF":0.0000,"publicationDate":"2022-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"9","resultStr":"{\"title\":\"On Assignment Problems Related to Gromov-Wasserstein Distances on the Real Line\",\"authors\":\"Robert Beinert, Cosmas Heiß, G. Steidl\",\"doi\":\"10.48550/arXiv.2205.09006\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Let $x_1<\\\\dots<x_n$ and $y_1<\\\\dots<y_n$, $n \\\\in \\\\mathbb N$, be real numbers. We show by an example that the assignment problem $$ \\\\max_{\\\\sigma \\\\in S_n} F_\\\\sigma(x,y) := \\\\frac12 \\\\sum_{i,k=1}^n |x_i - x_k|^\\\\alpha \\\\, |y_{\\\\sigma(i)} - y_{\\\\sigma(k)}|^\\\\alpha, \\\\quad \\\\alpha>0, $$ is in general neither solved by the identical permutation (id) nor the anti-identical permutation (a-id) if $n>2 +2^\\\\alpha$. Indeed the above maximum can be, depending on the number of points, arbitrary far away from $F_\\\\text{id}(x,y)$ and $F_\\\\text{a-id}(x,y)$. The motivation to deal with such assignment problems came from their relation to Gromov-Wasserstein divergences which have recently attained a lot of attention.\",\"PeriodicalId\":185319,\"journal\":{\"name\":\"SIAM J. Imaging Sci.\",\"volume\":\"24 1\",\"pages\":\"0\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2022-05-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"9\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"SIAM J. Imaging Sci.\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.48550/arXiv.2205.09006\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM J. Imaging Sci.","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.48550/arXiv.2205.09006","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
On Assignment Problems Related to Gromov-Wasserstein Distances on the Real Line
Let $x_1<\dots0, $$ is in general neither solved by the identical permutation (id) nor the anti-identical permutation (a-id) if $n>2 +2^\alpha$. Indeed the above maximum can be, depending on the number of points, arbitrary far away from $F_\text{id}(x,y)$ and $F_\text{a-id}(x,y)$. The motivation to deal with such assignment problems came from their relation to Gromov-Wasserstein divergences which have recently attained a lot of attention.
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