AI in Topological Data Analysis: Understanding High-Dimensional Data Structures
Murali Krishna Pasupuleti · 2024
Abstract: This book presents a rigorous, interdisciplinary investigation into the convergence of Artificial Intelligence (AI) and Topological Data Analysis (TDA) as a transformative framework for modeling and interpreting high-dimensional data structures. It addresses a fundamental challenge in modern data science: traditional statistical and machine learning techniques often struggle to preserve the global geometric and topological properties of complex datasets. By leveraging tools from algebraic topology—such as persistent homology, simplicial complexes, and Betti numbers—TDA enables the extraction of robust, multi-scale topological features from noisy, sparse, and nonlinear data. The book introduces a comprehensive framework in which topological descriptors are integrated into AI pipelines through persistence diagrams, barcodes, and vectorized representations. Methodologies include differentiable TDA layers, topological regularization in deep learning, manifold learning via Mapper and Reeb graphs, and Bayesian inference with topological priors. Applications span across domains including neuroscience, genomics, medical imaging, finance, and computer vision. Empirical results and case studies demonstrate how topology-aware AI models enhance robustness, reduce overfitting, and provide semantically meaningful representations of data. The book concludes by identifying open challenges—such as the scalability and differentiability of topological operations—and outlines a roadmap for future developments in topology-native machine learning. Through this synthesis, the work establishes TDA not only as a diagnostic tool but as a foundational principle for next-generation AI systems in high-dimensional data environments. Keywords Topological Data Analysis, Artificial Intelligence, Persistent Homology, High-Dimensional Data, Simplicial Complexes, Betti Numbers, Manifold Learning, Mapper Algorithm, Reeb Graphs, Dimensionality Reduction, Topological Priors, Differentiable TDA, Topological Regularization, Federated Learning, Bayesian Inference, Explainable AI, Algebraic Topology, Complex Systems, Geometric Machine Learning, Shape-Aware AI