Formation of machine learning features based on computational topology methods

The use of traditional methods of algebraic topology to obtain information about the shape of an object is associated with the problem of forming a small amount of information: Betti numbers and Euler characteristics. The central tool for topological data analysis is the persistent homology method, which summarizes the geometric and topological information in the data using persistent diagrams and barcodes. Based on persistent homology methods, analysis of topological data can be performed to obtain information about the shape of an object. The construction of persistent barcodes and persistent diagrams in computational topology does not allow one to construct a Hilbert space with a scalar product. The possibility of applying the methods of topological data analysis is based on the mapping of persistent diagrams into the Hilbert space; one of the ways of such mapping is the method of constructing a persistent landscape. Its advantages are that it is reversible, so it does not lose any information, and it has persistence properties. The paper considers mathematicalmodels and functions for representing persistent landscape objects based on the persistent homology method. Methods for converting persistent barcodes and persistent diagrams into persistent landscape functions are considered. Associated with persistent landscape functions is a persistent landscape kernel that forms a mapping into a Hilbert space with a dot product. A formula is proposed for determining the distance between persistent landscapes, which allows you to find the distance between images of objects. The persistent landscape functions map persistent diagrams to Hilbert space. Examples of determining the distance between images based on the construction of persistent landscape functions for these images are given. Representations of topological characteristics in various models of computational topology are considered. Extended results for single-parameter persistence modules to multi-parameter persistence modules.