Besov function approximation and binary classification on low-dimensional manifolds using convolutional residual networks

Hao Liu, Minshuo Chen, Tuo Zhao, Wenjing Liao*

*Corresponding author for this work

Research output: Chapter in book/report/conference proceedingConference proceedingpeer-review

10 Citations (Scopus)

Abstract

Most of existing statistical theories on deep neural networks have sample complexities cursed by the data dimension and therefore cannot well explain the empirical success of deep learning on high-dimensional data. To bridge this gap, we propose to exploit the low-dimensional structures of the real world datasets and establish theoretical guarantees of convolutional residual networks (ConvResNet) in terms of function approximation and statistical recovery for binary classification problem. Specifically, given the data lying on a d-dimensional manifold isometrically embedded in RD, we prove that if the network architecture is properly chosen, ConvResNets can (1) approximate {\it Besov functions} on manifolds with arbitrary accuracy, and (2) learn a classifier by minimizing the empirical logistic risk, which gives an {\it excess risk} in the order of n−(s/2s+2(s∨d)), where s is a smoothness parameter. This implies that the sample complexity depends on the intrinsic dimension d, instead of the data dimension D. Our results demonstrate that ConvResNets are adaptive to low-dimensional structures of data sets.
Original languageEnglish
Title of host publicationProceedings of the 38th International Conference on Machine Learning (ICML 2021)
EditorsMarina Meila, Tong Zhang
PublisherML Research Press
Pages6770-6780
Number of pages11
Publication statusPublished - 18 Jul 2021
Event38th International Conference on Machine Learning, ICML 2021 - Virtual
Duration: 18 Jul 202124 Jul 2021
https://icml.cc/virtual/2021/index.html
https://icml.cc/Conferences/2021
https://proceedings.mlr.press/v139/

Publication series

NameProceedings of Machine Learning Research
Volume139
ISSN (Print)2640-3498

Conference

Conference38th International Conference on Machine Learning, ICML 2021
Period18/07/2124/07/21
Internet address

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