In signal coding there are two famous theorems – Shannon Theorems, which put limits on coding efficiency in lossless and in lossy modes.
However, in Neural Networks there is Cybenko Theorem, which puts constraints on efficiency of Back-Propagation Neural Network as an approximation of a target function (notice that Neural Network is targeted to approximate functions).
In my words Cybenko Theorem states:
“Each continuous function defined over a compact domain can be approximated with any given precision by sigmoidial functions (or a set of sigmodial functions is dense within a set of continuous functions over a compact domain).”
Consequently if a given function is continuous then with sufficient large hidden layer the function can be approximated with any pre-defined precision.
However, if a given function has at least one discontinuity then back-propagation neural network may totally fails in approximation.
Notice that similar constraint is observed in Fourier Analysis: by means of the trigonometric functions (sin and cos) we can approximate any continuous function defined over a closed interval [a,b]. But if a function has a “jumpy” discontinuity then Gibbs effect appears around the discontinuity point.
23+ years’ programming and theoretical experience in the computer science fields such as video compression, media streaming and artificial intelligence (co-author of several papers and patents).
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