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Deep belief networks are exact

arXiv · AI, language, vision and robotics · article · Sep 4, 2026 · UTC

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.

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