Deep belief networks are exact

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.

Artificial Intelligence (cs.AI); Machine Learning (cs.LG); Probability (math.PR)

arXiv:2609.05572 [cs.AI]

(or arXiv:2609.05572v1 [cs.AI] for this version)

https://doi.org/10.48550/arXiv.2609.05572 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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.