Geometric Methods for Machine Learning and Optimization
summary
Hierarchical data embeds into low‑dimensional hyperbolic space because tree and hyperbolic volumes grow exponentially, so hyperbolic representations reduce required dimension; experiments show improved metrics and a working hyperbolic perceptron with recovered complexity bounds. Optimization over matrix-valued parameters moves to manifolds, with a Riemannian Frank‑Wolfe that yields sublinear convergence on geodesically convex sets but creates a harder linear oracle; the oracle is solvable for positive definite matrices with interval bounds and practical approximations work.
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chapters (8)
- 0:00Classically, Optimization and Machine Learning methods have assumed a Euclidean setting. However, many applications involve non-Euclidean data
- 0:45Overview
- 2:40Machine Learning in hyperbolic space
- 3:35Example: Classification
- 5:01Optimization on matrix manifolds
- 6:25Geodesic Convexity
- 6:57Riemannian Optimization
- 9:35Summary
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