  Despite significant advances in the field of deep learning in applications to
various fields, explaining the inner processes of deep learning models remains
an important and open question. The purpose of this article is to describe and
substantiate the geometric and topological view of the learning process of
neural networks. Our attention is focused on the internal representation of
neural networks and on the dynamics of changes in the topology and geometry of
the data manifold on different layers. We also propose a method for assessing
the generalizing ability of neural networks based on topological descriptors.
In this paper, we use the concepts of topological data analysis and intrinsic
dimension, and we present a wide range of experiments on different datasets and
different configurations of convolutional neural network architectures. In
addition, we consider the issue of the geometry of adversarial attacks in the
classification task and spoofing attacks on face recognition systems. Our work
is a contribution to the development of an important area of explainable and
interpretable AI through the example of computer vision.
