The convergence analysis of a problem solved by RGD on Stiefel.

[grandowife]

Hello,

There is a  Riemannian optimization problem defined on Stiefel that is solved by the  Riemmainnian Gradient Descent (RGD) and is proved to be converged by drawing the convergence curves.

Now I have to conduct the convergence analysis mathematically.

To this end, I have read many references about the convergence analysis of the regular GD algorithm for solving a problem defined in Euclidean, but I think that the problem in Euclidean and that in Riemannian should be treated differently.

Meanwhile, I have read the brilliant paper  "Global rates of convergence for nonconvex optimization on manifolds" written by Mr. Boumal.

It gives me great inspiration, however, since my major is not mathematical, I can not truly understand the theorems or the lemmas in this paper.

So now I am stuck in here, and I don't know what my next step is.

Hope that someone can give me a hint.

I could be a great help!

Thank you very much.

Best.

Shi.

[Nicolas Boumal]

Hello Shi,

This is a rather wide question: it is difficult to provide a helpful reply in a forum post. We may be able to help with a more precise question about an isolated issue.

Beyond that, if the main obstacle is to get to the point where you can be comfortable with the type of analyses that you found in my paper with P.-A. Absil and C. Cartis, then I would recommend that you read Chapters 3 and 4 in my book (nicolasboumal.net/book). If you skip the proofs and the exercises at first, and focus on the main definitions and the text surrounding the definitions that motivate them, then hopefully you will get a clearer picture of the important concepts.

Best wishes,

Nicolas