A Uniqueness Theorem for Exact Separable Log-Partition Decompositions

[2607.22710] A Uniqueness Theorem for Exact Separable Log-Partition Decompositions Skip to main content Search Submit Donate Log in Computer Science > Information Theory arXiv:2607.22710 (cs) [Submitted on 20 Jul 2026 ( v1 ), last revised 26 Aug 2026 (this version, v2)] Title: A Uniqueness Theorem for Exact Separable Log-Partition Decompositions Authors: Michael P. Rubin View PDF HTML (experimental) Abstract: Let $\mathcal{X}$ be a finite set, let $p$ and $Q$ be strictly positive probability distributions on $\mathcal{X}$, and let $u\in\mathbb{R}^{\mathcal{X}}$ be a log-weight. Define $\Psi_p(u)=\log\sum_x p(x)e^{u(x)}$ and $\pi_{p,u}(x)=p(x)e^{u(x)-\Psi_p(u)}$. The variational free energy $F(Q;p,u)=D(Q\|p)-\mathbb{E}_Q[u]$ satisfies the exact identity $\Psi_p(u)=-F(Q;p,u)+D(Q\|\pi_{p,u})$, whose residual is the canonical divergence of the dually flat simplex. We prove a converse. Suppose $G(Q;p,u)=A(Q,p)+B(Q,u)$ is additively separable and its gap $\Psi_p(u)+G(Q;p,u)$ is nonnegative and vanishes whenever $Q=\pi_{p,u}$. Then $G=F$ and the gap equals $D(Q\|\pi_{p,u})$; $A$ and $B$ are unique up to a $Q$-dependent additive gauge. The gap is not assumed to be a divergence or to depend only on the target, and no continuity, measurability, differentiability, convexity, or membership in a prescribed divergence family is assumed. The proof bounds increments of a corrected accuracy term by centered log-moment-generating costs of order $k^{-2}$ per step. Telescoping $k$ equal exponential tilts gives an $O(k^{-1})$ bound and forces every increment to vanish. Thus target factorization and strictness are conclusions. Counterexamples show that separability, nonnegativity, and tightness are each necessary. Within this class, no $\alpha$-divergence distinct from Kullback--Leibler and no Rényi divergence of order other than one can occur as the residual. Comments: 10 pages. Substantially revised and strengthened: removes the target-only and regularity assumptions, replaces the proof with a telescoping exponential-tilt argument, and expands the information-geometric framing and sharpness analysis. Prepared for submission to Information Geometry Subjects: Information Theory (cs.IT) ; Mathematical Physics (math-ph); Probability (math.PR) MSC classes: 62B11 (Primary) 53B12, 94A17, 39B22, 62F15, 82B03 (Secondary) Cite as: arXiv:2607.22710 [cs.IT] (or arXiv:2607.22710v2 [cs.IT] for this version) https://doi.org/10.48550/arXiv.2607.22710 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Michael P. Rubin [ view email ] [v1] Mon, 20 Jul 2026 18:57:34 UTC (12 KB) [v2] Wed, 26 Aug 2026 18:33:05 UTC (13 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Additional Features Audio Summary Current browse context: cs.IT < prev | next > new | recent | 2026-07 Change to browse by: cs math math-ph math.IT math.MP math.PR References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

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