{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/exact-fractional-inference-via-re","title":"Exact Fractional Inference via Re-Parametrization & Interpolation between Tree-Re-Weighted- and Belief Propagation- Algorithms","arxiv_id":"2301.10369","date":"2023-01-25","proceeding":null,"authors":["Hamidreza Behjoo","Michael Chertkov"],"abstract":"Computing the partition function, $Z$, of an Ising model over a graph of $N$ \\enquote{spins} is most likely exponential in $N$. Efficient variational methods, such as Belief Propagation (BP) and Tree Re-Weighted (TRW) algorithms, compute $Z$ approximately by minimizing the respective (BP- or TRW-) free energy. We generalize the variational scheme by building a $\\lambda$-fractional interpolation, $Z^{(\\lambda)}$, where $\\lambda=0$ and $\\lambda=1$ correspond to TRW- and BP-approximations, respectively. This fractional scheme -- coined Fractional Belief Propagation (FBP) -- guarantees that in the attractive (ferromagnetic) case $Z^{(TRW)} \\geq Z^{(\\lambda)} \\geq Z^{(BP)}$, and there exists a unique (\\enquote{exact}) $\\lambda_*$ such that $Z=Z^{(\\lambda_*)}$. Generalizing the re-parametrization approach of \\citep{wainwright_tree-based_2002} and the loop series approach of \\citep{chertkov_loop_2006}, we show how to express $Z$ as a product, $\\forall \\lambda:\\ Z=Z^{(\\lambda)}{\\tilde Z}^{(\\lambda)}$, where the multiplicative correction, ${\\tilde Z}^{(\\lambda)}$, is an expectation over a node-independent probability distribution built from node-wise fractional marginals. Our theoretical analysis is complemented by extensive experiments with models from Ising ensembles over planar and random graphs of medium and large sizes. Our empirical study yields a number of interesting observations, such as the ability to estimate ${\\tilde Z}^{(\\lambda)}$ with $O(N^{2::4})$ fractional samples and suppression of variation in $\\lambda_*$ estimates with an increase in $N$ for instances from a particular random Ising ensemble, where $[2::4]$ indicates a range from $2$ to $4$. We also discuss the applicability of this approach to the problem of image de-noising.","url_abs":"https://arxiv.org/abs/2301.10369v4","url_pdf":"https://arxiv.org/pdf/2301.10369v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"exact-fractional-inference-via-re","repo_url":"https://github.com/hamidrezabehjoo/fractional-trw","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}