Deep belief networks are exact

2026-09-04

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.

Problem

• deep belief networks
• sigmoid belief network
• probability distribution
• Brouwer's fixed-point theorem

Innovation

• sigmoid belief network
• exact representation
• Brouwer's fixed-point theorem