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I have this simple neural networks ("Self Organizing Maps"):

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What is the purpose of the lateral connections beetween the perceptrons (neurons) ?

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Lateral connections exist so that the update of a neuron forces the neighboring neurons also to be updated but to a lesser degree.

Think of each neuron that has n inputs as a vector, a point in an n-dimensional space. It's coordinates are the values of the weights on its inputs.

When the network receives an input - also an n-dimensional vector, each neuron calculates the distance between its weights and the input vector. The neuron whose weights are the closest to the input is the winner, and is allowed to update its weights. It updates them by moving one step towards the input vector. The size of the step is equal to the learning rate.

While it moves, it uses the lateral connections to pull its neighbors (or push some of them away, depending on their distance to the winner and on the form of the function that you use for the lateral connections). As a result, the neighbors also move but they move less than the winner.

Only the winner and its neighbors are allowed to update their weights. All other neurons don't move. The neighbors are those who are closer to the winner than a certain radius. This radius is a hyper-parameter.

This type of learning allows to chart the space of the input vectors. The result is the weights of the neurons are distributed more or less uniformly in the set of the input vectors. After training, each neuron can be thought of as a representative of some region of the input space.

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