A review of Topological Data Analysis and Machine Learning

Abstract

During the past decade, seeing how computational topology has injected some of the core ideas and principles from algebraic and differential topology into applications has been a huge success. This fusion of theories gave birth to a new field named Topological Data Analysis (TDA), which has significant value in various fields, ranging from computational biology to personalized medicine and dynamic data analysis. Going beyond its foundational applications, TDA has enriched and complemented classical machine and deep learning frameworks in establishing what is now known as "topological machine learning." In this paper, we review the present landscape of this emerging field, emphasizing how it merges with machine learning algorithms such as deep neural networks. Each method confers special advantages, targeting areas like machine learning integration, network reconstruction, classification of network regimes, or reduction of noisy data. We describe common methodologies, discuss current implementations, and anticipate future challenges in topological machine learning.