Geometric Foundations of AI (I): Parameter Spaces, Statistical Manifolds, and Optimization

How should we think geometrically about the parameters of a modern learning system? When is the Fisher matrix a genuine metric, what does natural gradient really make invariant, and what replaces smooth Riemannian geometry when parameterizations become redundant or singular? Geometric Foundations of AI (I) develops a rigorous, graduate-level framework for answering these questions. Beginning with statistical manifolds and information geometry, it moves through Fisher-Rao geometry, natural-gradient methods, variational inference, Legendre and Bregman duality, high-dimensional convex geometry, symmetry and quotient structure, loss-landscape geometry, singular learning theory, parameter-space topology, and spectral questions at modern scale. Throughout, the emphasis is not only on what a theorem says, but on the hypotheses that make it true and on what survives when the theorem is transferred to neural networks. Designed for graduate students and researchers in machine learning, statistics, applied mathematics, and optimization, the book combines classical geometric ideas with contemporary AI questions. Worked examples, carefully separated levels of evidence, and tiered exercises help readers distinguish intrinsic structure from coordinate artifacts-and develop a more reliable geometric understanding of optimization and learning.