Ribbons from Independence Structure: Hypercontractivity, $Φ$-Mutual Information, and Matrix $Φ$-Entropy
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917223760658432 |
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| author | Wang, Chenyu Gohari, Amin |
| author_facet | Wang, Chenyu Gohari, Amin |
| contents | We study the hypercontractivity ribbon and the $Φ$-ribbon for joint distributions that obey a given independence structure, obtaining tight bounds in some basic regimes. For general independence structures, modeled as a hypergraph whose hyperedges specify mutually independent subcollections of random variables, we provide an explicit inner bound on the $Φ$-ribbon described by a simple convex hull of incidence vectors. We also provide a new multipartite generalization version and a $Φ$-mutual information analogue of the Zhang--Yeung inequality, which implies nontrivial points in the hypercontractivity ribbon and the $Φ$-ribbon respectively. Finally, we propose the matrix $Φ$-ribbon based on matrix $Φ$-entropy and establish the tensorization and data processing properties, together with the calculation of an exact matrix SDPI constant for the doubly symmetric binary source. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_18516 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Ribbons from Independence Structure: Hypercontractivity, $Φ$-Mutual Information, and Matrix $Φ$-Entropy Wang, Chenyu Gohari, Amin Information Theory We study the hypercontractivity ribbon and the $Φ$-ribbon for joint distributions that obey a given independence structure, obtaining tight bounds in some basic regimes. For general independence structures, modeled as a hypergraph whose hyperedges specify mutually independent subcollections of random variables, we provide an explicit inner bound on the $Φ$-ribbon described by a simple convex hull of incidence vectors. We also provide a new multipartite generalization version and a $Φ$-mutual information analogue of the Zhang--Yeung inequality, which implies nontrivial points in the hypercontractivity ribbon and the $Φ$-ribbon respectively. Finally, we propose the matrix $Φ$-ribbon based on matrix $Φ$-entropy and establish the tensorization and data processing properties, together with the calculation of an exact matrix SDPI constant for the doubly symmetric binary source. |
| title | Ribbons from Independence Structure: Hypercontractivity, $Φ$-Mutual Information, and Matrix $Φ$-Entropy |
| topic | Information Theory |
| url | https://arxiv.org/abs/2601.18516 |