On overcoming the Curse of Dimensionality in Neural Networks

09/02/2018

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by   Karen Yeressian, et al.

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Let H be a reproducing Kernel Hilbert space. For i=1,...,N, let x_i∈R^d and y_i∈R^m comprise our dataset. Let f^*∈ H be the unique global minimiser of the functional J(f) = 1/2 f_H^2 + 1/N∑_i=1^N1/2 f(x_i)-y_i^2. In this paper we show that for each n∈N there exists a two layer network where the first layer has nm number of basis functions Φ_x_i_k,j for i_1,...,i_n∈{1,...,N}, j=1,...,m and the second layer takes a weighted summation of the first layer, such that the functions f_n realised by these networks satisfy f_n-f^*_H≤ O(1/√(n))for all n∈N. Thus the error rate is independent of input dimension d, output dimension m and data size N.