Computational difficulties of multidimensional tasks, called the ``curse of dimensionality’’, have long been known. On the other hand, almost deterministic behaviour of some randomized models and algorithms depending on large numbers of variables can be attributed to the ``blessing of dimensionality’’. These phenomena can be explained by rather counter-intuitive properties of geometry of high-dimensional spaces. They imply concentration of values of sufficiently smooth functions of many variables around their mean values.

In the lecture, it will be shown how these properties of high-dimensional geometry can be employed to obtain some insights into suitability of various types of neural networks for classification of large data sets. Probabilistic bounds on network complexity will be derived using concentration properties of approximation errors based on Azuma and McDiarmid inequlities. Consequences for choice of network architectures will be analyzed in terms of growth functions and VC dimensions of sets of network input-output functions. General results will be illustrated by examples of deep perceptron networks with various piecewise polynomial activation functions (ReLU, RePU). Probabilistic results will be complemented by some concrete constructions. Connections with the central paradox of coding theory and pseudo-noise sequences will be discussed.