Theory of deep convolutional neural networks II : Spherical analysis
Research output: Journal Publications and Reviews (RGC: 21, 22, 62) › 21_Publication in refereed journal › peer-review
|Journal / Publication||Neural Networks|
|Online published||6 Aug 2020|
|Publication status||Published - Nov 2020|
|Link to Scopus||https://www.scopus.com/record/display.uri?eid=2-s2.0-85089067956&origin=recordpage|
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 Sd−1 of ℝd. Our analysis presents rates of uniform approximation when the approximated function lies in the Sobolev space W r∞(Sd−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.
- Approximation theory, Convolutional neural networks, Deep learning, Sobolev spaces, Spherical analysis