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I am learning about SOM and also about Hopfield networks. I yet don't understand why I would use SOM over Hopfield and vice versa.

What kind of problems I should resolve with SOM and which with Hopfield?

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I've worked with SOMs, but not with Hopfield networks. A quick glance at Wikipedia says that Hopfield networks are used for associative memory, while SOMs are used for dimensionality reduction.

So SOMs are used to visualize high-dimensional data and for clustering high-dimensional data, if the data actually lies on a low (2D usually) manifold in a high-dimensional space. SOMs incorporate a (usually) 2D neighbor-defining grid which is how it does what it does, and at least according to Wikipedia this feature is unique among ANNs.

I guess Hopfield networks could be used for some kind of classification, but their main use would be something more like a robust, approximate value-value kind of lookup. Someone who knows them better could give a better answer.

EDIT: To correct the points addressed by alto's comment.

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Since Hopfield networks are autoassociative it's more useful to think of them as a value-value lookup rather than a key-value lookup, which would be heteroassociative. They can be used for denoising, filling in missing variables, or even approximately solving hard optimization problems (like TSP). I also wouldn't say SOMs are for classification (unless you would say k-means is for classification, which I definitely would not). – alto Jul 16 '12 at 21:04

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