Analysis of convolutional neural network image classifiers in a hierarchical max-pooling model with additional local pooling

In practical applications of image classification, methods based on convolutional neural networks are used as standard nowadays. However, the success these is not explained sufficiently from a theoretical point view. For this, statistical model for namely hierarchical max-pooling with additional local pooling, introduced. Here pooling enables to combine parts which have variable relative distance towards each other. Various network classifiers introduced and analyzed in view their rate convergence misclassification risk estimates optimal risk. This analysis provides explanation why general architectures that include some kind layers useful classification situations gives hints choosing right architecture. Furthermore, finite sample size performance illustrated by applying them simulated real data.