# Difference between revisions of "Oja's rule"

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[[File:ArtificialNeuronModel english.png|thumb|right|400px|Model of a neuron. <em>j</em> is the index of the neuron when there is more than one neuron. For a linear neuron, the activation function is not present (or simply the identity function).]] | [[File:ArtificialNeuronModel english.png|thumb|right|400px|Model of a neuron. <em>j</em> is the index of the neuron when there is more than one neuron. For a linear neuron, the activation function is not present (or simply the identity function).]] | ||

− | As with Hebb's rule, we use a linear neuron. Given a set of k-dimensional inputs represented as a column vector [[File:Hebb1.png]], and a linear neuron with (initially random) synaptic weights from the inputs [[File:Hebb2.png]] the output the neuron is defined as follows: | + | As with Hebb's rule, we use a linear neuron. Given a set of k-dimensional inputs represented as a column vector [[File:Hebb1.png]], and a linear neuron with (initially random) [[Synapse|synaptic]] weights from the inputs [[File:Hebb2.png]] the output the neuron is defined as follows: |

[[File:Hebb3.png|center]] | [[File:Hebb3.png|center]] |

## Revision as of 18:08, 21 July 2014

**Oja's rule**, developed by Finnish computer scientist Erkki Oja in 1982, is a stable version of Hebb's rule.^{[1]}

## Model

As with Hebb's rule, we use a linear neuron. Given a set of k-dimensional inputs represented as a column vectorError creating thumbnail: Unable to save thumbnail to destination

, and a linear neuron with (initially random) synaptic weights from the inputs Error creating thumbnail: Unable to save thumbnail to destination

the output the neuron is defined as follows:
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Oja's rule gives the update rule which is applied after an input pattern is presented:

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ignored for n>1 since η is small.
It can be shown that Oja's rule extracts the first principal component of the data set. If there are many Oja's rule neurons, then all will converge to the same principal component, which is not useful. Sanger's rule was formulated to get around this issue.

## References

- ↑ Oja, Erkki (November 1982). "Simplified neuron model as a principal component analyzer".
*Journal of Mathematical Biology***15**(3): 267–273