Geometry for Machine Learning With Engineering Applications: Theory, Examples, and Python Implementations (Computational Mathematics Library)

A rigorous bridge between modern differential geometry and deployable machine learning systems. This graduate-level text unifies Riemannian optimization, Lie theory, and equivariant representations into a coherent toolkit for building stable, symmetry-aware learning algorithms. It is written for readers who want proofs and implementation details side-by-side—and who intend to use both.What sets this book apartMathematically rigorous and implementation-focused: from definitions and theorems to practical algorithms that run on GPUs.Unified geometric perspective: curvature, connections, group actions, and spectrum inform both optimization landscapes and model architectures.Engineering-grade case studies: robotics (SE(3) pose), signal processing (Grassmann subspaces), medical imaging (SPD statistics), molecular modeling (E(3)-equivariant networks), vision on spheres (SO(3) harmonics), and more.Designed for research and production: emphasizes numerical stability, invariances, and reproducibility.Structure of every chapterTheory: precise definitions, lemmas, propositions, and theorems; geometric intuition without sacrificing rigor.Examples: worked case studies connecting the chapter’s mathematics to real engineering tasks.Python Implementation: reference code illustrating core abstractions (manifolds, metrics, exp/log, retractions, transports), optimizers, and equivariant layers; integrates cleanly with NumPy/PyTorch/JAX workflows.What you will learnDifferential and Riemannian geometry for ML: metrics, connections, geodesics, curvature, Jacobi fields, and comparison geometry—how they shape optimization and generalization.Matrix manifolds and quotients: Stiefel/Grassmann, SPD cones, fixed-rank geometries; submersions, homogeneous spaces, and symmetry reduction for identifiability and efficiency.Lie groups and representation theory: exponential/log, adjoint actions, Haar integration; Peter–Weyl, characters, and intertwiners for designing equivariant layers.Equivariant deep learning: steerable convolutions on groups and homogeneous spaces; gauge-equivariant networks on meshes and surfaces; sampling and quadrature on manifolds.Optimization on manifolds: first- and second-order methods, stochastic/online variants, proximal and splitting schemes, trust-region and quasi-Newton, with convergence guarantees.Probabilistic and statistical geometry: information geometry and natural gradients; Wasserstein/Otto calculus; diffusion/score models and HMC on manifolds and Lie groups.Spectral and hyperbolic geometry: Laplace–Beltrami operators, heat kernels, Hodge theory, nonpositive curvature, and hyperbolic embeddings for hierarchical data.Software practice: manifold-aware autodiff, numerically safe exp/log and transports, differentiating through eigendecompositions, and testing equivariance. Read more