riemannian-geometry
4 free lessons tagged riemannian-geometry across Math. Each one is a short sequence of focused steps with narration and a five-question quiz at the end — take them in any order, no signup required.
Applications, Geodesic Convexity, and Why Non-Convex Can Still Be Solvable
Where this machinery earns its place: low-rank matrix completion, synchronization, and PCA. Plus the two ideas that explain why non-convex manifold problems are often solved to global optimality anyway.
Retractions and Riemannian Algorithms
How to move along a curved space without solving differential equations: retractions as cheap approximations to geodesics, vector transport, and the Riemannian versions of gradient descent, conjugate gradients, and trust regions.
Tangent Spaces, Metrics, and the Riemannian Gradient
Building the machinery: the linear space of allowed directions at a point, the inner product that gives it geometry, and why the Riemannian gradient is the ambient gradient projected rather than a new derivative.
When Your Parameters Live on a Curved Space
Rotations, subspaces, low-rank matrices and covariances are not vectors in flat space. Why treating those constraints as penalties or projections wastes structure, and what it means to say the search space is a manifold.