Number of hidden layers in a neural network model

Would someone be able to explain to me or point me to some resources of why (or situations where) more than one hidden layer would be necessary or useful in a neu开发者_如何转开发ral network?

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Basically more layers allow more functions to be represented. The standard book for AI courses, "Artificial Intelligence, A Modern Approach" by Russell and Norvig, goes into some detail of why multiple layers matter in Chapter 20.

One important point is that with a sufficiently large single hidden layer, you can represent every continuous function, but you will need at least 2 layers to be able to represent every discontinuous function.

In practice, though, a single layer is enough at least 99% of the time.

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1. That's more similar to the way the brain works (which might not necessarily be a computational advantage, but a lot of people are researching NN to gain insight about the way the mind works, rather than to solve real world problems.
2. Its easier to achieve some kinds of invariance using more layers. For example, an image classifier that works regardless of where in the image the object is found, or the object's size. see Bouvrie, J. , L. Rosasco, and T. Poggio. "On Invariance in Hierarchical Models". Advances in Neural Information Processing Systems (NIPS) 22, 2009.

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Each layer effectively raises the potential "complexity" of adaptation in an exponential fashion (as opposed to a multiplicative fashion of adding more nodes to a single layer).