Mathematics / research
Optimization on manifolds
Translate optimization algorithms to curved constraint spaces using tangent geometry, retractions, and intrinsic second-order models.
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Module 01 research
Optimization on manifolds
Translate optimization algorithms to curved constraint spaces using tangent geometry, retractions, and intrinsic second-order models.
- 01
Embedded constraints and tangent projections
Derive feasible first-order directions from regular constraints and compute projections only in the metric for which they are valid.
31 min - 02
Riemannian gradients and metric choice
Recover a gradient from a differential using the chosen metric and recognize metric design as structured preconditioning.
32 min - 03
Retractions, vector transports and exponential maps
Separate feasible first-order updates from geodesic motion and check the length behavior of transported tangent vectors.
33 min - 04
Riemannian Hessians and second-order conditions
Differentiate the gradient covariantly and retain the curvature correction that an ambient Hessian alone misses.
33 min - 05
Geodesic convexity and curvature effects
State convexity on a declared geodesic domain and distinguish intrinsic convexity from ambient coordinate convexity.
33 min - 06
Optimization on spheres and Stiefel manifolds
Keep orthonormal frames feasible and distinguish the induced and canonical Stiefel metrics before computing gradients or steps.
34 min - 07
Grassmann geometry and subspace learning
Optimize a subspace rather than a labeled frame and express stationarity through a basis-invariant eigenvector residual.
34 min - 08
Quotient manifolds and gauge redundancy
Remove equivalent parameterizations with a regular group action and ensure that the metric and objective descend consistently.
35 min - 09
Riemannian trust-region algorithms
Compare a tangent quadratic model with actual retracted decrease and state the model and subproblem conditions needed for convergence.
35 min - 10
Stochastic Riemannian optimization and rates
State unbiased tangent-noise and retraction-regularity assumptions before deriving finite-horizon stationary-point bounds.
36 min