Neural Network Frames on the Sphere

We construct a multiresolution analysis of the standard Hilbert space on a Euclidean sphere, which can be implemented directly by neural networks. The neural networks may utilize any sufficiently smooth function as an activation function, and their size can be determined in advance. We define frame operators that can analyze data selected at scattered sites, rather than any particular set of points. The number of vanishing moments increases with the order of the frames. In particular, the frames can detect discontinuities in arbitrarily high order derivatives. The neural networks do not require any training in the traditional sense. INTRODUCTION Many recent applications require the modelling and analysis of signals whose domain is the surface of a sphere. Examples include the study of seismic signals, gravitational phenomenon, hydrogen atom, solar corona, and medical imaging of the brain. An important tool in this study, analogous to Fourier analysis in classical signal processing, consists of spherical harmonics. Although spherical harmonics have been studied by mathematicians since the 1780s, there are persistent difficulties in practical applications; for example, a proper choice of the sites where the samples should be taken. Recently, we have developed [3, 4] techniques to obviate the need to choose the samples at any particular sites. At the cost of oversampling, our techniques enable us to model arbitrary continuous functions on the sphere with arbitrary accuracy The research of HNM and JDW was supported, in part, by grants DMS-9971846 and DMS-9971276 from the National Science Foundation. The research of FJN and JDW was supported, in part, by grant F49620-98-1-0204 from the U. S. Air Force Office of Scientific Research. using samples collected at “scattered sites”. In [5], we have also constructed polynomial frames suitable for analysing functions based on such scattered samples. In this paper, we describe a construction of frames using neural networks for the analysis of functions on the sphere. We observe that there are some efforts to approximate standard wavelets using neural networks [2], and also to construct neural networks based on certain wavelet-like functions as activation function [1]. A novelty of our paper is to construct neural networks which are themselves frames, which in addition, have an increasing number of vanishing moments, and utilize standard activation functions. In the next section, we develop certain notations, and review some facts about spherical harmonics. In Section 3, we describe the construction of neural network frames, and in Section 4, we give an example of how these can be utilized to detect the location of certain “tornados”. SPHERICAL HARMONICS Spherical Harmonics Let q ≥ 1 be an integer which will be fixed throughout the rest of this paper, and let S be the (surface of the) unit sphere in the Euclidean space R, with dμq being its usual volume element. We note that the volume element is invariant under arbitrary coordinate changes. The volume of S is ωq := ∫ Sq dμq = 2π Γ((q + 1)/2) . (2.1) Corresponding to dμq, we have the inner product and L (S) norms, 〈f, g〉Sq := ∫ Sq f(x)g(x)dμq(x), ‖f‖Sq,2 = 〈f, g〉. (2.2) The class of all measurable functions f : S → C for which ‖f‖Sq, 2 < ∞ will be denoted by L(S), with the usual understanding that functions that are equal almost everywhere are considered equal as elements of L(S). All continuous complex valued functions on S will be denoted by C(S). For integer l ≥ 0, the restriction to S of a homogeneous harmonic polynomial of degree l is called a spherical harmonic of degree l. Most of the following information is based on [7] and [10, §IV.2], although we use a different notation. The class of all spherical harmonics of degree l will be denoted by Hql , and the class of all spherical harmonics of degree l ≤ n will be denoted by Πn. Of course, Π q n = ⊕n l=0 H q l , and it comprises the restriction to S of all algebraic polynomials in q+1 variables of total degree not exceeding n. The dimension of Hql is given by d ql := dimH q l = 