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I skimmed it, but at a glance, isn't this almost identical to the Taylor Series? https://en.wikipedia.org/wiki/Taylor_series https://en.wikipedia.org/wiki/Tayl
by personjerry 8y ago
I skimmed it, but at a glance, isn't this almost identical to the Taylor Series?
https://en.wikipedia.org/wiki/Taylor_series https://en.wikipedia.org/wiki/Taylor_series
- YjSe2GMQ 8y agoWell, in the sense that polynomials are universal approximators too (on a compact domain). See this comment: https://news.ycombinator.com/item?id=19709834 https://news.ycombinator.com/item?id=19709834
- tprice7 8y agoThe Stone-Weierstrass theorem is not related to Taylor series, in fact the target function need not have a derivative, let alone have all higher-order derivatives, a prerequisite for the Taylor series to even exist.
- tprice7 8y agoNo.
- Too 8y agoFourier transforms also smell quite similar. With those you also know how many samples you need to represent a function which changes at some known maximum frequency, using the Nyquist theorem. Is there anything translatable with neural nets? Like would you be able to prove you need X nr of neurons to represent a given function? It seems the choice of function inside the neuron is kindof arbitary or is it locked to the sigmoid function? If so you could put cos and i*sin as function in your neurons and your neural net essentially becomes DFT? https://en.wikipedia.org/wiki/Discrete_Fourier_transform https://en.wikipedia.org/wiki/Discrete_Fourier_transform
- DavidSJ 8y agoTaylor series only approximate an (infinitely differentiable) function well locally, i.e. in some neighborhood of the approximation point. There are other methods applicable to less smooth functions and which approximate well uniformly (bounding the maximum absolute error) or by some other metric (e.g. L2 distance — integrated squared error).