Analysis of convolutional neural network image classifiers in a rotationally symmetric model

TL;DR

This paper analyzes the convergence rate of convolutional neural network classifiers for images modeled as functions with rotational symmetry, demonstrating their ability to overcome the curse of dimensionality under certain conditions.

Contribution

It provides a theoretical framework showing CNN classifiers can bypass the curse of dimensionality in symmetric image models, with finite sample analysis included.

Findings

CNN classifiers achieve fast convergence under symmetry assumptions.

The approach circumvents the curse of dimensionality in binary image classification.

Finite sample behavior is validated with simulated and real data.

Abstract

Convolutional neural network image classifiers are defined and the rate of convergence of the misclassification risk of the estimates towards the optimal misclassification risk is analyzed. Here we consider images as random variables with values in some functional space, where we only observe discrete samples as function values on some finite grid. Under suitable structural and smoothness assumptions on the functional a posteriori probability, which includes some kind of symmetry against rotation of subparts of the input image, it is shown that least squares plug-in classifiers based on convolutional neural networks are able to circumvent the curse of dimensionality in binary image classification if we neglect a resolution-dependent error term. The finite sample size behavior of the classifier is analyzed by applying it to simulated and real data.