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I have trained a neural network using MATLAB and am ready to deploy it into my software. Right now, I include it in my software by programming the structure of the neural network and the connection weights and doing some matrix multiplication.

But there has to be a better way to do this: neural networks are basically linear separators. This means that it should be possible to model the outputs as some linear function of the inputs.

So here's the question: given the structure of a neural network and the connection weights after training, how would I come up with the correct linear function of the inputs that models the functionality of the network?

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It would seem to depend on the complexity of your neural network. A single perceptron is a linear separator. On the other hand, it is known that multilayer feed-forward neural networks are universal approximators (see here: http://dl.acm.org/citation.cfm?id=70408 ): i.e. they can approximate any function to arbitrary accuracy. Obviously for such a network, you should not expect to be able to express its outputs as a linear function of its inputs.

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  • $\begingroup$ Sure, they can approximate functions to arbitrary accuracy. But the approximation that they learn is ultimately also a function, yes? So wouldn't that mean that I could write a function `f(x1, x2, ..., xn) -> (y1, y2, ..., yn) that perfectly models this approximation? $\endgroup$ – inspectorG4dget Mar 8 '12 at 16:19
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    $\begingroup$ Of course -- a neural network represents a function. But not necessarily a linear function. $\endgroup$ – Aaron Roth Mar 8 '12 at 16:31
  • $\begingroup$ Please correct me if I'm wrong, but do you mean "not a linear function IN 2 DIMENSIONS"? It's my understanding that it's always a linear function, possibly, though in higher dimensional spaces, which would make it super-linear in 2-D space $\endgroup$ – inspectorG4dget Mar 8 '12 at 16:42
  • $\begingroup$ Incidentally, if you don't apply any sigmoid functions (ie. all neurons are linear) then any multilayer perceptron is also a linear function (every layer is a linear function, so you're just chaining linear functions). The point is that if your NN is a linear function, then you may as well be training a linear function (a single layer NN without sigmoid), or better yet, just calculate the optimal solution directly with least squares. What you can do is compute each layer as a linear function and apply the sigmoid to the resulting vector. $\endgroup$ – Peter Mar 8 '12 at 16:46

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