Lp APPROXIMATION CAPABILITIES OF SUM-OF-PRODUCT AND SIGMA-PI-SIGMA NEURAL NETWORKS

This paper studies the L(p) approximation capabilities of sum-of-product (SOPNN) and sigma-pi-sigma (SPSNN) neural networks. It is proved that the set of functions that are generated by the SOPNN with its activation function in Lloc;p(R) is dense in L;p(K) for any compact set K⊂ R;N, if and only if the activation function is not a polynomial almost everywhere. It is also shown that if the activation function of the SPSNN is in Lloc;∞(R), then the functions generated by the SPSNN are dense in L;p(K) if and only if the activation function is not a constant (a.e.).