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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.12227 |
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| _version_ | 1866915291793981440 |
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| author | Ma, Ya-Jing Wang, Feng Wu, Xian-Yuan Cai, Kai-Yuan |
| author_facet | Ma, Ya-Jing Wang, Feng Wu, Xian-Yuan Cai, Kai-Yuan |
| contents | Given probability distributions ${\bf p}=(p_1,p_2,\ldots,p_m)$ and ${\bf q}=(q_1,q_2,\ldots, q_n)$ with $m,n\geq 2$, denote by ${\cal C}(\bf p,q)$ the set of all couplings of $\bf p,q$, a convex subset of $\R^{mn}$. Denote by ${\cal C}_e({\bf p},{\bf q})$ the finite set of all extreme points of ${\cal C}(\bf p,q)$. It is well known that, as a strictly concave function, the Shannan entropy $H$ on ${\cal C}(\bf p,q)$ takes its minimal value in ${\cal C}_e({\bf p},{\bf q})$. In this paper, first, the detailed structure of ${\cal C}_e({\bf p},{\bf q})$ is well specified and all extreme points are enumerated by a special algorithm. As an application, the exact solution of the minimum-entropy coupling problem is obtained. Second, it is proved that for any strict Schur-concave function $Ψ$ on ${\cal C}(\bf p,q)$, $Ψ$ also takes its minimal value on ${\cal C}_e({\bf p},{\bf q})$. As an application, the exact solution of the minimum-entropy coupling problem is obtained for $(Φ,\hbar)$-entropy, a large class of entropy including Shannon entropy, Rényi entropy and Tsallis entropy etc. Finally, all the above are generalized to multi-marginal case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_12227 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An Explicit Description of Extreme Points of the Set of Couplings with Given Marginals: with Application to Minimum-Entropy Coupling Problems Ma, Ya-Jing Wang, Feng Wu, Xian-Yuan Cai, Kai-Yuan Probability Given probability distributions ${\bf p}=(p_1,p_2,\ldots,p_m)$ and ${\bf q}=(q_1,q_2,\ldots, q_n)$ with $m,n\geq 2$, denote by ${\cal C}(\bf p,q)$ the set of all couplings of $\bf p,q$, a convex subset of $\R^{mn}$. Denote by ${\cal C}_e({\bf p},{\bf q})$ the finite set of all extreme points of ${\cal C}(\bf p,q)$. It is well known that, as a strictly concave function, the Shannan entropy $H$ on ${\cal C}(\bf p,q)$ takes its minimal value in ${\cal C}_e({\bf p},{\bf q})$. In this paper, first, the detailed structure of ${\cal C}_e({\bf p},{\bf q})$ is well specified and all extreme points are enumerated by a special algorithm. As an application, the exact solution of the minimum-entropy coupling problem is obtained. Second, it is proved that for any strict Schur-concave function $Ψ$ on ${\cal C}(\bf p,q)$, $Ψ$ also takes its minimal value on ${\cal C}_e({\bf p},{\bf q})$. As an application, the exact solution of the minimum-entropy coupling problem is obtained for $(Φ,\hbar)$-entropy, a large class of entropy including Shannon entropy, Rényi entropy and Tsallis entropy etc. Finally, all the above are generalized to multi-marginal case. |
| title | An Explicit Description of Extreme Points of the Set of Couplings with Given Marginals: with Application to Minimum-Entropy Coupling Problems |
| topic | Probability |
| url | https://arxiv.org/abs/2505.12227 |