Mathematical foundation of High-Dimensional Data Analysis: Leveraging Topology and Geometry for Enhanced Model Interpretability in AI

Abstract

One of the most important challenges for modern AI and machine learning is the analysis of high-dimensional data. Traditional methods face serious complications in such cases due to high complexity of datasets: the curse of dimensionality, overfitting, and lack of transparency of model behavior. In this paper, we adopt a novel approach to analyze high-dimensional data; topological and geometric techniques will be exploited, taking advantage of better model interpretability and deeper insights into the structure. Precisely, we discuss Topological Data Analysis, mainly Persistent Homology  (Edelsbrunner et al., 2002), which allows the extraction of topological features-like loops and connected components that enable the extracting knowledge about the global structure of data. We also see how some concepts of differential geometry and Riemannian geometry (Do Carmo, 1976) can be used to cast light on manifold data structure lying at the heart of any attempt at modeling intrinsic patterns in high-dimensional spaces.

We will review how these mathematical pillars, combined with state-of-the-art techniques for dimensionality reduction like t-SNE, UMAP, Principal Component Analysis, are able to provide interpretable and low-dimensional representations of high-dimensional data that can be used to understand models and make decisions. Case studies are also included, which explain the practical working of these methods in AI systems and show how much complex models can be made transparent using these, especially in domains that are very critical, such as healthcare  (Caruana et al., 2015), finance (Chen et al., 2018), and autonomous systems ( Wang et al., 2019).

We also discuss some of the difficulties in using these methods for practical applications: computational complexity; the need for large-scale data processing (Bengio et al., 2007); and integration of topological and geometric intuition with the rest of the machine learning pipeline (Zhu et al., 2020). We conclude with possible future directions of research toward fine-tuning these methods and exploring their broader applicability to AI in its quest for more robust, interpretable, and reliable AI models. Given this work, we focus on how linking topology, geometry, and AI bears great promise for solving one of today's critical challenges: model interpretability in high-dimensional data analysis.