mlsanityData to 7 Oct 2026

Theory of Deep Convolutional Neural Networks II: Spherical Analysis

arXiv 2007.14285 ↗First posted 28 Jul 2020cs.LG math.FA stat.MLcited by 22

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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.

Key facts. Cited by 22 papers in the map (data through 7 Oct 2026). First posted on arXiv 28 Jul 2020.

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Citing papers per year: 2021: 3; 2022: 2; 2023: 7; 2024: 2; 2025: 3; 2026: 5.

22 papers.

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  1. 2021Besov Function Approximation and Binary Classification on Low-Dimensional Manifolds Using Convolutional Residual Networkscited by 32
  2. 2021Theory of Deep Convolutional Neural Networks III: Approximating Radial Functionscited by 22
  3. 2023Optimal rates of approximation by shallow ReLUk neural networks and applications to nonparametric regressioncited by 15
  4. 2023Rates of Approximation by ReLU Shallow Neural Networkscited by 12
  5. 2023The Hessian perspective into the Nature of Convolutional Neural Networkscited by 10
  6. 2024On the rates of convergence for learning with convolutional neural networkscited by 10
  7. 2023Expressivity and Approximation Properties of Deep Neural Networks with ReLUk Activationcited by 8
  8. 2022Approximation analysis of CNNs from a feature extraction viewcited by 6
  9. 2021Non-asymptotic Excess Risk Bounds for Classification with Deep Convolutional Neural Networkscited by 4
  10. 2023Exploring the Complexity of Deep Neural Networks through Functional Equivalencecited by 4
  11. 2025Higher Order Approximation Rates for ReLU CNNs in Korobov Spacescited by 4
  12. 2023Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networkscited by 3
10 more
  1. 2024Spherical Analysis of Learning Nonlinear Functionalscited by 3
  2. 2025DFNN: A Deep Fr'echet Neural Network Framework for Learning Metric-Space-Valued Responsescited by 2
  3. 2026Learning Sparse Compositional Functions with Norm-Constrained Neural Networkscited by 2
  4. 2022Optimal Convergence Rates of Deep Convolutional Neural Networks: Additive Ridge Functionscited by 1
  5. 2023Optimal Approximation and Learning Rates for Deep Convolutional Neural Networkscited by 1
  6. 2025Adaptive Classification of Interval-Valued Time Seriescited by 1
  7. 2026Sparse-Aware Neural Networks for Nonlinear Functionals: Mitigating the Exponential Dependence on Dimensioncited by 0
  8. 2026Simultaneous CNN Approximation on Manifolds with Applications to Boundary Value Problemscited by 0
  9. 2026New Fractional Ambiguity Function Integrated with CNN-Based Machine Learning for Signal Classificationcited by 0
  10. 2026Optimal Neural Network Approximation via Empirical Least Squares with Deterministic Samplescited by 0