Notes for "Why does deep and cheap learning work so well?" (ArXiv:1608.08225v1/cond-mat.dis-nn) by Lin and Tegmark.

SimpleNeuralNets.ipynb

ps - small typo, in the paragraph right after eq. (11) it says that the approximation is exact as lambda -> \infty. It should be "as lambda -> 0."

BTW @hwlin76, for the multivariate case, are you thinking of representing it as nested networks, e.g. g(u,v,w) = uvw as g = f(u, f(v,w)) where f(u,v) = uv, or do you have a similar (explicit) construction to the paper for the 3-variable case, with a 3-n-1 network? I imagine this should be possible/easy, but I haven't tried yet figuring out what the nx3 matrix would be, nor what \mu would become. And if that's the case, do you have the inductive result for the p-term multivariate product?

Before I dive into the multivariate analysis, it would be good to know which way you're thinking of it, and if you have these constructive results ready then we could formulate the implementation that way directly.