Riemannian Gradient Descent: Convergence Theorem and Line Search Method

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Riemannian connections: What they are and why we care

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Symmetry Property: Riemannian Connection in Geometry

Explores symmetries, Riemannian connection, vector fields, and Lie bracket in geometry.

Riemannian metrics and gradients: Computing gradients from extensions

Explores computing gradients on Riemannian manifolds through extensions and retractions, emphasizing orthogonal projectors and smooth extensions.

Differentiating Vector Fields: How Not to Do It

Discusses the challenges in differentiating vector fields on submanifolds and the importance of choosing the right method.

Differentiating vector fields: Why do it?

Explores the importance of differentiating vector fields and the correct methodology to achieve it, emphasizing the significance of going beyond the first order.

Newton's method: Optimization on manifolds

Explores Newton's method for optimizing functions on manifolds using second-order information and discusses its drawbacks and fixes.

Gradients on Riemannian submanifolds, local frames

Discusses gradients on Riemannian submanifolds and the construction of local frames.

All things Riemannian: metrics, (sub)manifolds and gradients

Covers the definition of retraction, open submanifolds, local defining functions, tangent spaces, and Riemannian metrics.

Choosing a Step Size

Explores choosing a step size in optimization on manifolds, including backtracking line-search and the Armijo method.

Riemannian metrics and gradients: Riemannian gradients

Explains Riemannian submanifolds, metrics, and gradients computation on manifolds.