# Theory of Deep Convolutional Neural Networks II: Spherical Analysis

arXiv: https://arxiv.org/abs/2007.14285 · HTML: https://mlsanity.com/paper/2007.14285/

## Key facts

- Cited by 22 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jul 2020.

## Built on by (most cited first, 22 of 22)

- [Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networks](https://mlsanity.com/paper/2109.02832/) (2021)
- [Theory of Deep Convolutional Neural Networks III: Approximating Radial Functions](https://mlsanity.com/paper/2107.00896/) (2021)
- [Optimal rates of approximation by shallow ReLUk neural networks and applications to nonparametric regression](https://mlsanity.com/paper/2304.01561/) (2023)
- [Rates of Approximation by ReLU Shallow Neural Networks](https://mlsanity.com/paper/2307.12461/) (2023)
- [The Hessian perspective into the Nature of Convolutional Neural Networks](https://mlsanity.com/paper/2305.09088/) (2023)
- [On the rates of convergence for learning with convolutional neural networks](https://mlsanity.com/paper/2403.16459/) (2024)
- [Expressivity and Approximation Properties of Deep Neural Networks with ReLUk Activation](https://mlsanity.com/paper/2312.16483/) (2023)
- [Approximation analysis of CNNs from a feature extraction view](https://mlsanity.com/paper/2210.09041/) (2022)
- [Non-asymptotic Excess Risk Bounds for Classification with Deep Convolutional Neural Networks](https://mlsanity.com/paper/2105.00292/) (2021)
- [Exploring the Complexity of Deep Neural Networks through Functional Equivalence](https://mlsanity.com/paper/2305.11417/) (2023)
- [Spherical Analysis of Learning Nonlinear Functionals](https://mlsanity.com/paper/2410.01047/) (2024)
- [Higher Order Approximation Rates for ReLU CNNs in Korobov Spaces](https://mlsanity.com/paper/2501.11275/) (2025)
- [Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks](https://mlsanity.com/paper/2308.09605/) (2023)
- [Learning Sparse Compositional Functions with Norm-Constrained Neural Networks](https://mlsanity.com/paper/2605.25608/) (2026)
- [DFNN: A Deep Fr'echet Neural Network Framework for Learning Metric-Space-Valued Responses](https://mlsanity.com/paper/2510.17072/) (2025)
- [Optimal Convergence Rates of Deep Convolutional Neural Networks: Additive Ridge Functions](https://mlsanity.com/paper/2202.12119/) (2022)
- [Optimal Approximation and Learning Rates for Deep Convolutional Neural Networks](https://mlsanity.com/paper/2308.03259/) (2023)
- [Adaptive Classification of Interval-Valued Time Series](https://mlsanity.com/paper/2504.03318/) (2025)
- [Sparse-Aware Neural Networks for Nonlinear Functionals: Mitigating the Exponential Dependence on Dimension](https://mlsanity.com/paper/2604.06774/) (2026)
- [Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problems](https://mlsanity.com/paper/2605.04126/) (2026)
- [New Fractional Ambiguity Function Integrated with CNN-Based Machine Learning for Signal Classification](https://mlsanity.com/paper/2606.08110/) (2026)
- [Optimal Neural Network Approximation via Empirical Least Squares with Deterministic Samples](https://mlsanity.com/paper/2608.06687/) (2026)

## Abstract (arXiv)

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 mathbbSd-1 of ℝd. Our analysis presents rates of uniform approximation when the approximated function lies in the Sobolev space Wr_∞ (mathbbSd-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.