Deviation theorems for solutions of differential equations and applications to lower bounds on parallel complexity of sigmoids (Q1338215)

The main result (see also the above abstract) of the paper states that if two different functions \(f_1\), \(f_2\) are computed by means of sigmoids with the parallel complexity \(d\), then the difference \(|f_1- f_2|\) grows not slower than \((\exp^{(d)}(p))^{- 1}\) (and not faster than \(\exp^{(d)}(p))\), where \(\exp^{(d)}\) is the \(d\) times iteration of the exponential function and \(p\) is a certain polynomial, thus one cannot rather good approximate \(f_1\) with a precise parallel complexity \(d\) by means of a function \(f_2\) with a less parallel complexity. The number of zeros in the intervals of a function computed by sigmoids is estimated and all the obtained bounds are sharp.