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I can imagine a lot of math's theorems and laws, but I can not imagine the "process" universal approximation theorem is talking about.

Can you explain in simple words the proof of universal approximation theorem?

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    $\begingroup$ In Wikipedia's terminology, you can approximately expand a continuous real-valued function on the unit hypercube into a finite linear combination of functions of the form $\varphi \circ a_i$ where $a_i$ are affine functions and $\varphi$ can be taken to be a single, rather "nice", function. If my limited understanding is correct, these $a_i$ form the "hidden layer" of the network. And yes, this expansion of a "generic" function into a sum of "nice" functions is similar to expansion into power series, Fourier series, or wavelet series. $\endgroup$
    – Ian
    Commented Mar 20, 2017 at 12:31

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The article A visual proof that neural nets can compute any function (by Michael Nielsen) gives a nice geometric intuition.

If you're interested, you can find the proof easily around, by following the Wikipedia references.

The linked book is generally good for understanding the basics of neural networks.

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  • $\begingroup$ ..What chapter? $\endgroup$
    – MadHatter
    Commented Oct 28, 2017 at 15:51

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