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Here is a thought of the day: in general a principal geodesic component does not pass through the intrinsic mean on the manifold. Which is not the case in the usual Euclidean PCA. The reference is the article by Stephan Huckemann. I think it is worth to carefully go through it.

With the help of my friend I have discovered the whole submanifold of people, writing on the stuff I am currently interested. The topic being geodesic PCA (or PGA). I was going through all the references from our groups (Johns Hopkins, Utah, Paris etc groups) and digging. Then a friend forwarded me a paper, which uncovered the whole trove of papers that I haven’t discovered yet. There is a lot of mutual linking in that new submanifold, but very few links cross over to the subset of papers I was reading initially. Hence the difficulty in finding.

Of course, a google scholar search of “PGA” would have revealed them eventually. And in retrospect one can say “aha! here is the reference! how did you miss it?” But I guess before the AI is invented one has to carefully parse through all possible links on the topic. We are lucky to have google. It is difficult to imagine how one would search for an article before all the digitization.