AI's Quiet Geometry: Riemannian and Manifold Learnings (lightcapai.medium.com)

This piece is a beginner-friendly explainer showing how two geometry-driven ideas—manifold learning and Riemannian optimization—are coming together in modern ML. Manifold learning looks for low-dimensional “shapes” hidden inside high-dimensional data (think t-SNE, Isomap, UMAP extracting a curved 2D sheet from thousands of pixel values). Riemannian optimization is the complementary technique for training models while respecting those shapes: instead of taking straight-line (Euclidean) steps, optimizers move along the manifold using geometric notions like tangent spaces, Riemannian gradients and retractions so parameters never leave the allowed surface. The convergence matters because many real-world datasets and model constraints are inherently non‑Euclidean (spheres, hyperbolic spaces for hierarchies, rotation manifolds in robotics). Training directly on the discovered manifold preserves topology and distances, yields more stable and interpretable solutions, and can improve convergence and generalization versus forcing problems into flat Euclidean space or using ad‑hoc constraints. Practically, this means metric-aware optimizers and manifold-aware embeddings (e.g., hyperbolic graph embeddings) are increasingly useful in vision, representation learning, graph models and control. In short: finding the shape of your data and optimizing on that shape reduces hacks, aligns models with reality, and unlocks better performance and theoretical guarantees.