A visual proof that neural nets can compute any function

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One of the most striking facts about neural networks is that they can compute any function at all. That is, suppose someone hands you some complicated, wiggly function, $f(x)$: No matter what the function, there is guaranteed to be a neural network so that for every possible input, $x$, the value $f(x)$ (or some close approximation) is output from the network, e.g.: This result holds even if the function has many inputs, $f = f(x_1, \ldots, x_m)$, and many outputs. For instance, here's a network computing a function with $m = 3$ inputs and $n = 2$ outputs: This result tells us that neural…

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