  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.
