# What are Boltzmann machines used for?

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- How are Boltzmann machines trained?
- What is the energy function good for?

A **Boltzmann machine** is a stochastic artificial neural network developed by Geoffrey Hinton and Terrence J. Sejnowski in 1985.^{[1]} These networks are named after the Boltzmann distribution. Boltzmann machines with no connection restrictions are very difficult to train. However, if the connections between the neurons are restricted, the learning process can be greatly simplified, which means that Limited Boltzmann machines can be used to solve practical problems.

### Table of Contents

### construction

A Boltzmann machine, like a Hopfield network, is a network of neurons in which an energy level is defined. As in Hopfield networks, the neurons only accept binary values (0 or 1), but in contrast behave stochastically. The energy level \ ({\ displaystyle E} \) of a Boltzmann machine is defined as in a Hopfield network:

- \ ({\ displaystyle E = - \ left (\ sum _ {i

where:

- \ ({\ displaystyle w_ {ij}} \) is the weight of the connection between neuron \ ({\ displaystyle i} \) and \ ({\ displaystyle j} \).
- \ ({\ displaystyle s_ {i}} \) is the state \ ({\ displaystyle s_ {i} \ in \ {0,1 \}} \) of the neuron \ ({\ displaystyle i} \).
- \ ({\ displaystyle \ theta _ {i}} \) is the threshold value of a neuron \ ({\ displaystyle i} \). (\ ({\ displaystyle - \ theta _ {i}} \) is the value from which a neuron is activated.)

The connections of a Boltzmann machine have two limitations:

- \ ({\ displaystyle w_ {ii} = 0 \ qquad \ forall i} \). (No neuron has a connection with itself.)
- \ ({\ displaystyle w_ {ij} = w_ {ji} \ qquad \ forall i, j} \). (All connections are symmetrical.)

The weightings can be represented in the form of a symmetrical matrix \ ({\ displaystyle W} \), the main diagonal of which consists of zeros.

Just as with the Hopfield network, the Boltzmann machine tends to reduce the value of the energy defined in this way with successive updates, i.e. ultimately to minimize it until a stable state is reached.

### Restricted Boltzmann machine

A so-called Restricted Boltzmann Machine (RBM) consists of visible units and hidden units. The feature vector is applied to the invisible units.

The "restricted" comes from the fact that the visible units are not connected to each other and the hidden units are not connected to each other. However, the visible units are fully connected to the hidden units. So they form a bipartite, undirected graph. This is illustrated below:

The parameters to be learned are the weights of the edges between visible and hidden units and the bias vectors \ ({\ displaystyle b_ {h}, b_ {v}} \) of the hidden and visible units. These are learned using the Contrastive Divergence Algorithm.^{[2]}

Restricted Boltzmann Machines were used for collaborative filtering on Netflix.^{[3]}

### Web links

### Individual evidence

- ↑ David H. Ackley, Geoffrey E. Hinton, Terrence J. Sejnowski:
*A Learning Algorithm for Boltzmann Machines.*In:*Cognitive science*, Volume 9, Issue 1, January 1985, pp. 147-169. On Wiley.com (PDF, English), accessed on February 13, 2021, doi: 10.1207 / s15516709cog0901_7. - ↑ Geoffrey Hinton: A practical guide to training restricted Boltzmann machines. 2010.
- ↑ Ruslan Salakhutdinov, Andriy Mnih, Geoffrey Hinton: Restricted Boltzmann machines for collaborative filtering. In: Proceedings of the 24th international conference on machine learning. 2007, pp. 791-798.

**Categories:**Neuroinformatics | Computational Neuroscience | Machine learning | Ludwig Boltzmann

**Status of information:**04/27/2021 2:01:26 AM CEST

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