Topological Methods in Machine Learning and Data Analysis: A Mathematical Perspective

Abstract

Topology, as a part of mathematics that studies the properties of space that are invariant under continuous transformations, has come into focus of the learning community as one of the effective approaches to addressing the difficulties connected with the analysis of the learning matter. This paper examines key concepts of topological approaches – persistent homology and the mapper algorithm to discover the theoretical background of topological methods, their goals and purposes, and their use in analyzing the patterns and structures of the data, improving the machine learning algorithms, and searching for valuable patterns in large datasets. This work shows how topology can be effectively used to solve important problems of the modern data science, including data in high dimensions, model interpretability and detecting small changes and outliers.