Geometry of the Nonlinear Jacobian Chain (The Methematical Model of ResNet and Transformers)
This project builds a geometric lens for modern deep learning by studying the Riemannian structures generated by Jacobian chains in nonlinear compositional models. The central object is the depth-wise product of layer Jacobians, which induces a pullback metric on data/feature space and encodes how local volumes, directions, and curvature are transported through the network. We ask when depth preserves near-isometry, when it creates anisotropy or singularities, and how training reshapes these geometric invariants. The goal is a principled bridge between depth, curvature/spectra, stability, and practical trainability—yielding measurable diagnostics and theory-guided design principles.
