Note that the gradient ∇F (X i ) can be naturally considered as a matrix in R n×m .
point X i + ∇F (X i ) leaves the manifold M d , because its rank is generally greater than d. why Riemannian optimization methods The standard Riemannian gradient method first projects the gradient step onto the tangent space at the current point X i and then retracts it back to the manifold: where R is the retraction operator, and p T M is the projection onto the tangent space.