When does a mixture of products contain a product of mixtures?

Guido F. Montúfar, Jason Morton

Research output: Contribution to conferencePaperpeer-review


We derive relations between theoretical properties of restricted Boltzmann machines (RBMs), popular machine learning models which form the building blocks of deep learning models, and several natural notions from discrete mathematics and convex geometry. We give implications and equivalences relating RBM-representable probability distributions, perfectly reconstructible inputs, Hamming modes, zonotopes and zonosets, point configurations in hyperplane arrangements, linear threshold codes, and multi-covering numbers of hypercubes. As a motivating application, we prove results on the relative representational power of mixtures of product distributions and products of mixtures of pairs of product distributions (RBMs) that formally justify widely held intuitions about distributed representations. In particular, we show that a mixture of products requiring an exponentially larger number of parameters is needed to represent the probability distributions which can be obtained as products of mixtures.

Original languageEnglish (US)
StatePublished - 2013
Event1st International Conference on Learning Representations, ICLR 2013 - Scottsdale, United States
Duration: May 2 2013May 4 2013


Conference1st International Conference on Learning Representations, ICLR 2013
Country/TerritoryUnited States

All Science Journal Classification (ASJC) codes

  • Education
  • Computer Science Applications
  • Linguistics and Language
  • Language and Linguistics


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