Approximating smooth functions by deep neural networks with sigmoid activation function
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We study the power of deep neural networks (DNNs) with sigmoid activation function. Recently, it was shown that DNNs approximate any d-dimensional, smooth function on a compact set with a rate of order W^-p/d, where W is the number of nonzero weights in the network and p is the smoothness of the function. Unfortunately, these rates only hold for a special class of sparsely connected DNNs. We ask ourselves if we can show the same approximation rate for a simpler and more general class, i.e., DNNs which are only defined by its width and depth. In this article we show that DNNs with fixed depth and a width of order M^d achieve an approximation rate of M^-2p. As a conclusion we quantitatively characterize the approximation power of DNNs in terms of the overall weights W_0 in the network and show an approximation rate of W_0^-p/d. This more general result finally helps us to understand which network topology guarantees a special target accuracy.
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