Title :
Simultaneous Lp-approximations of polynomials and derivatives on the whole space
Author_Institution :
Aichi-Gakuin Univ., Aichi, Japan
Abstract :
We have obtained a sufficient condition that a linear sum of an activation function can simultaneously approximate polynomials and their derivatives in the sense of Lp(Rd, μ). If the probability measure μ is rapidly decreasing, a wide variety of differentiable functions satisfy this condition. For the Gaussian measure, rapidly increasing functions such as et, exp(t2 /2) and others can be activation functions. The proof is constructive and elementary, which enables the approximation formulas to be written explicitly
Keywords :
polynomial approximation; Gaussian measure; activation function linear sum; activation functions; derivative Lp-approximations; differentiable functions; neural nets; polynomial Lp-approximations; probability measure; simultaneous Lp-approximations;
Conference_Titel :
Artificial Neural Networks, 1999. ICANN 99. Ninth International Conference on (Conf. Publ. No. 470)
Conference_Location :
Edinburgh
Print_ISBN :
0-85296-721-7
DOI :
10.1049/cp:19991173