### Abstract

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 multicovering 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 language | English (US) |
---|---|

Pages (from-to) | 321-347 |

Number of pages | 27 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 29 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 2015 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*29*(1), 321-347. https://doi.org/10.1137/140957081

}

*SIAM Journal on Discrete Mathematics*, vol. 29, no. 1, pp. 321-347. https://doi.org/10.1137/140957081

**When does a mixture of products contain a product of mixtures?** / Montúfar, Guido F.; Morton, Jason.

Research output: Contribution to journal › Article

TY - JOUR

T1 - When does a mixture of products contain a product of mixtures?

AU - Montúfar, Guido F.

AU - Morton, Jason

PY - 2015/1/1

Y1 - 2015/1/1

N2 - 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 multicovering 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.

AB - 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 multicovering 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.

UR - http://www.scopus.com/inward/record.url?scp=84925348229&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84925348229&partnerID=8YFLogxK

U2 - 10.1137/140957081

DO - 10.1137/140957081

M3 - Article

AN - SCOPUS:84925348229

VL - 29

SP - 321

EP - 347

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 1

ER -