The essential rate of approximation for radial function manifold

Abstract

In this paper, we investigate the radial function manifolds generated by a linear combination of radial functions. Let W r p (B d ) be the usual Sobolev class of functions on the unit ball B d . We study the deviation from the radial function manifolds to W r p (B d ). Our results show that the upper and lower bounds of approximation by a linear combination of radial functions are asymptotically identical. We also find that the radial function manifolds and ridge function manifolds generated by a linear combination of ridge functions possess the same rate of approximation.

Almost optimal estimates for approximation and learning by radial basis function networks

“…The relationship between RBFN and polynomial approximation has already studied in Maiorov (2003) and Lin et al (2011a). In Maiorov (2003), Maiorov studied the approximation properties of the radial function manifold G N whose elements take the form as…”

“…They proved that the approximation capability of G N is not worse than that of polynomials of degrees at most n provided N ≥ (2d Lin et al (2011a) deduced that if the target function is radial and N ∼ n d−1 , then the approximation rate of polynomials is also not slower than that of G N . Noting that the utilized approximants in (6) are linear combinations of different univariate functions, it is difficult to determine the capacity of G N .…”

“…A series of important estimates for approximating functions in W r p by RBFNs were deduced in Bejancu (1997Bejancu ( , 2000, Johnson (1998), Lin et al (2011aLin et al ( , 2011b, Maiorov (2003Maiorov ( , 2005, Schaback (1995Schaback ( , 1996 and Xie and Cao (2013). For more details, Shcakback (1995Shcakback ( , 1996 gave the upper bound error for approximation by RBFNs with the wellknown plate spline activation function when the target function belongs to W r 2 .…”

Learning Algorithm of Neural Networks on Spherical Cap

“…In [12] [13] [14] [15], the following feed-forward networks defined on the unit sphere were considered and some approximation properties were studied:…”