Examining artificial intelligence algorithms with differential geometry

This thesis presents a systematic investigation of manifold learning algorithms through the lens of differential geometry and offers a unified geometric framework that connects the geodesic-based Isomap method with the spectral properties of the Laplace–Beltrami operator. The study first introduces the foundational concepts of Riemannian geometry–including metric tensors, geodesics, energy functionals, and the Laplace–Beltrami operator–and then provides a rigorous differential geometric interpretation of Isomap and Laplacian Eigenmaps. A key contribution of this work is the demonstration, for the first time in the literature, that these two widely used nonlinear dimensionality reduction techniques can be formulated under a common energy functional. In addition, a curvature-sensitive neighborhood selection strategy and a geodesic–spectral consistency metric are proposed. The experimental analyses conducted on classical synthetic manifolds such as the Swiss roll and S-curve are complemented by a geometric examination of embedding spaces produced by large language models (LLMs). The results reveal that LLM embeddings exhibit locally linear yet globally curved manifold behavior. Overall, this thesis provides an original perspective that bridges differential geometry with contemporary artificial intelligence, offering a coherent mathematical framework for understanding the intrinsic structure of high-dimensional data. Keywords: Manifold learning, differential geometry, Isomap, Laplace–Beltrami operator, dimensionality reduction, artificial intelligence.