2020/07/28 by Zhiying Fang, Fang, Zhiying, Han Feng +7 · 2 citations
Computer Science · Engineering · Mathematics · Physics and Astronomy · #Advanced Numerical Analysis Techniques #FOS: Computer and information sciences #FOS: Mathematics #Functional Analysis (math.FA) #Image and Signal Denoising Methods #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Model Reduction and Neural Networks #cs.LG #math.FA #stat.ML
paper · pdf · doi:10.48550/arxiv.2007.14285
arxiv created 2020/07/28 · openalex publication_date 2020/07/28 · arxiv updated 2020/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.