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

Topology and geometry of data manifold in deep learning