Formation of Machine Learning Features Based on the Construction of Tropical Functions

Abstract

One of the main methods of computational topology and topological data analysis is persistent homology, which combines geometric and topological information about an object using persistent diagrams and barcodes. The persistent homology method from computational topology provides a balance between reducing the data dimension and characterizing the internal structure of an object. Combining machine learning and persistent homology is hampered by topological representations of data, distance metrics, and representation of data objects. The paper considers mathematical models and functions for representing persistent landscape objects based on the persistent homology method. The persistent landscape functions allow you to map persistent diagrams to Hilbert space. The representations of topological functions in various machine learning models are considered. An example of finding the distance between images based on the construction of persistent landscape functions is given. Based on the algebra of polynomials in the barcode space, which are used as coordinates, the distances in the barcode space are determined by comparing intervals from one barcode to another and calculating penalties. For these purposes, tropical functions are used that take into account the basic structure of the barcode space. Methods for constructing rational tropical functions are considered. An example of finding the distance between images based on the construction of tropical functions is given. To increase the variety of parameters (machine learning features), filtering of object scanning by rows from left to right and scanning by columns from bottom to top are built. This adds spatial information to topological information. The method of constructing persistent landscapes is compatible with the approach of constructing tropical rational functions when obtaining persistent homologies.