Spherical Neural Networks Research Articles

Approximation by spherical neural networks with zonal functions

We address the construction and approximation for feed-forward neural networks (FNNs) with zonal functions on the unit sphere. The filtered de la VallA©e-Poussin operator and the spherical quadrature formula are used to construct the spherical FNNs. In particular, the upper and lower bounds of approximation errors by the FNNs are estimated, where the best polynomial approximation of a spherical function is used as a measure of approximation error. doi:10.1017/S1446181117000104

Simultaneous approximation by spherical neural networks

Approximation capabilities of the spherical neural networks (SNNs) are considered in this paper. Based on a known Taylor formula, we prove that, for non-polynomial target function, rates of simultaneously approximating the function itself and its (Laplace–Beltrami) derivatives by SNNs is not slower than those by the spherical polynomials (SPs). Then, the simultaneous approximation rates of SPs automatically derive the rates of SNNs.

Jackson-type inequalities for spherical neural networks with doubling weights

Recently, the spherical data processing has emerged in many applications and attracted a lot of attention. Among all the methods for dealing with the spherical data, the spherical neural networks (SNNs) method has been recognized as a very efficient tool due to SNNs possess both good approximation capability and spacial localization property. For better localized approximant, weighted approximation should be considered since different areas of the sphere may play different roles in the approximation process. In this paper, using the minimal Riesz energy points and the spherical cap average operator, we first construct a class of well-localized SNNs with a bounded sigmoidal activation function, and then study their approximation capabilities. More specifically, we establish a Jackson-type error estimate for the weighted SNNs approximation in the metric of Lp space for the well developed doubling weights.

Error Estimate for Spherical Neural Networks Interpolation

We present a type of spherical neural network (SNN) with bounded sigmoidal activation function and study its interpolation capability. We find that the provided SNN can exactly interpolate the training samples. Furthermore, based on the special structure of the presented SNN, we can bound the interpolation error by the modulus of smoothness of the target function, which is different from the previous results on the spherical scattered data interpolation problem.

The estimate for approximation error of spherical neural networks

Compared with planar hyperplane, fitting data on the sphere has been an important and an active issue in geoscience, metrology, brain imaging, and so on. In this paper, with the help of the Jackson-type theorem of polynomial approximation on the sphere, we construct spherical feed-forward neural networks to approximate the continuous function defined on the sphere. As a metric, the modulus of smoothness of spherical function is used to measure the error of the approximation, and a Jackson-type theorem on the approximation is established. Copyright © 2011 John Wiley & Sons, Ltd.

Essential rate for approximation by spherical neural networks

We consider the optimal rate of approximation by single hidden feed-forward neural networks on the unit sphere. It is proved that there exists a neural network with n neurons, and an analytic, strictly increasing, sigmoidal activation function such that the deviation of a Sobolev class W²(2r)(S(d)) from the class of neural networks Φ(n)(ϕ), behaves asymptotically as n(-2r/d-1). Namely, we prove that the essential rate of approximation by spherical neural networks is n(-2r/d-1).

Uncertainty principle in non-quantum “spherical modification” of the Hopfield model interacting with classical physical reality

In this paper a new approach to the problem of observation in quantum theory is suggested. The observing system is formalized as a “spherical neural network”. In those conditions the presence of an uncertainty principle depends on the mode of description of reality by the observing system.