Nonlinear Algebraic Topology: A Framework for High-Dimensional Data Analysis

Abstract

High-dimensional data analysis using NLAT shows a new impulse in the methods of algebraic topology which reveal specific features of topological structures connected with connected components, holes, and voids. This work aims to explore TDA methodologies, with focus on persistent homology, for solving a number of issues arising in the course of working with large, high-dimensional data across various disciplines including biology, neuroscience, and machine learning. While compared with the traditional linear methods of dimensionality reduction, which generally leads to the loss of structural information, the AL employs algebraic topology for nonlinear structure-preserving, making it feasible for multi-scale and multi-dimensional pattern analysis.

In this paper, we provide a detailed description of how to perform TDA for high-dimensional data and describe computational packages such as Ripser and Dipha, which are built for high-dimensional data. Furthermore, we analyze the opportunity for combining TDA with machine learning models for the improvement of classification, clustering, and anomaly detection. These techniques are illustrated by genomics and neuroscience instances, but talks raise issues of future computation, especially scalability and real-time.

Possible avenues for future work include creating interfaces to TDA tools that are even easier to use, integrating TDA technology into more common data analytics software, and exploring new areas of science and engineering such as quantum computing and cryptography. This work demonstrates that nonlinear algebraic topology is now moving as a powerful approach for high-dimensional data analysis and displaying that it is an essential instrument for scientists and engineers who searching for hidden structures of a large number of data in a variety of scientific and engineering fields.