Theory of Deep Convolutional Neural Networks II: Spherical Analysis

TL;DR

This paper provides a theoretical analysis of deep convolutional neural networks' ability to approximate functions on the sphere, establishing approximation rates and verifying their modeling capacity using spherical harmonic kernels.

Contribution

It introduces a spherical analysis framework for deep CNNs, deriving approximation rates for functions on the sphere and confirming their theoretical modeling capabilities.

Findings

Established uniform approximation rates for functions on the sphere.

Verified the modeling ability of deep CNNs with specific architectures.

Utilized spherical harmonic kernels to analyze convolutional filters.

Abstract

Deep learning based on deep neural networks of various structures and architectures has been powerful in many practical applications, but it lacks enough theoretical verifications. In this paper, we consider a family of deep convolutional neural networks applied to approximate functions on the unit sphere $\mathbb{S}^{d-1}$ of $\mathbb{R}^d$. Our analysis presents rates of uniform approximation when the approximated function lies in the Sobolev space $W^r_\infty (\mathbb{S}^{d-1})$ with $r>0$ or takes an additive ridge form. Our work verifies theoretically the modelling and approximation ability of deep convolutional neural networks followed by downsampling and one fully connected layer or two. The key idea of our spherical analysis is to use the inner product form of the reproducing kernels of the spaces of spherical harmonics and then to apply convolutional factorizations of filters to realize the generated linear features.