CNNs Avoid Curse of Dimensionality by Learning on Patches

TL;DR

This paper proposes a patch-based theoretical framework explaining CNNs' ability to generalize well in image classification, attributing it to their operation on image patches which mitigates the curse of dimensionality.

Contribution

It introduces the first a priori error bound for CNN generalization error based on patch processing, supported by quantitative and qualitative evidence.

Findings

CNNs operate on image patches, reducing effective dimensionality.

Patch-based theory explains effectiveness of data augmentation techniques.

Provides the first a priori generalization error bound for CNNs.

Abstract

Despite the success of convolutional neural networks (CNNs) in numerous computer vision tasks and their extraordinary generalization performances, several attempts to predict the generalization errors of CNNs have only been limited to a posteriori analyses thus far. A priori theories explaining the generalization performances of deep neural networks have mostly ignored the convolutionality aspect and do not specify why CNNs are able to seemingly overcome curse of dimensionality on computer vision tasks like image classification where the image dimensions are in thousands. Our work attempts to explain the generalization performance of CNNs on image classification under the hypothesis that CNNs operate on the domain of image patches. Ours is the first work we are aware of to derive an a priori error bound for the generalization error of CNNs and we present both quantitative and qualitative evidences in the support of our theory. Our patch-based theory also offers explanation for why data augmentation techniques like Cutout, CutMix and random cropping are effective in improving the generalization error of CNNs.