Deep belief networks are exact

A paper submitted to arXiv on 4 Sep 2026 proves that every strictly positive probability distribution on {-1,1}^n is represented exactly by a sigmoid belief network with finite parameters, answering a question posed by Ilya Sutskever and Geoffrey Hinton. The proof upgrades the pair's earlier probability-sharing approximation to an exact representation using Brouwer's fixed-point theorem.

Computer Science Artificial Intelligence Submitted on 4 Sep 2026 Title:Deep belief networks are exact View PDF /pdf/2609.05572 HTML experimental https://arxiv.org/html/2609.05572v1 Abstract:We prove that every strictly positive probability distribution on $\{-1,1\}^n$ is represented exactly by a sigmoid belief network with finite parameters. This answers a question of Sutskever and Hinton. The proof upgrades their probability-sharing approximation to exact representation using Brouwer's fixed-point theorem. Current browse context: cs.AI References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer What is the Explorer? https://info.arxiv.org/labs/showcase.html arxiv-bibliographic-explorer Connected Papers What is Connected Papers? https://www.connectedpapers.com/about Litmaps What is Litmaps? https://www.litmaps.co/ scite Smart Citations What are Smart Citations? https://www.scite.ai/ Code, Data and Media Associated with this Article alphaXiv What is alphaXiv? https://alphaxiv.org/ CatalyzeX Code Finder for Papers What is CatalyzeX? https://www.catalyzex.com DagsHub What is DagsHub? https://dagshub.com/ Gotit.pub What is GotitPub? http://gotit.pub/faq Hugging Face What is Huggingface? https://huggingface.co/huggingface ScienceCast What is ScienceCast? https://sciencecast.org/welcome Demos Recommenders and Search Tools Influence Flower What are Influence Flowers? https://influencemap.cmlab.dev/ CORE Recommender What is CORE? https://core.ac.uk/services/recommender 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 https://info.arxiv.org/labs/index.html .