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I know that mean curvature and diffusion-type flows are common in manifold learning because of their smoothing effects. I haven't seen Ricci flow used as much. Given that Ricci and diffusion-type flows behave quite similarly, what is the reason for this (e.g. computational)?

I was thinking that a discrete Ricci flow applied to graphs would have a similar effect to diffusion maps on graphs. I'm not as familiar with Ricci flows, so I don't have an intuition on the geometric behavior of Ricci vs diffusion flows on a graph.

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My limited exposure to manifold learning suggests that it is often searching for a submanifold in a high dimensional Euclidean space. There is one property that makes Ricci flow difficult for deforming submanifolds, which is that it is an intrinsic flow, and does not depend on the embedding. On the other hands, something like mean curvature flow is extrinsic, and depends in the embedding in a very explicit way.

In practice, what this means is that if you have a submanifold of a high dimensional Euclidean space, it's reasonably easy to come up with a numeric scheme which approximates mean curvature flow. In other words, you move your points a little bit in the direction of the mean curvature. To deform a submanifold by Ricci flow, you would first need to find a scheme to approximate Ricci flow (which has been done), but then find a way to embed your deformed surface back into your original space which is "close" to your original embedding. The latter is a nontrivial problem, and it's not clear to me that it can always be done (although if the codimension is large enough, Nash's embedding theorem can help).

There have been a fair number of papers using Ricci flow for computer vision and other applications, but I'm not sure how these works relate to manifold learning. It certainly seems like it might be possible to use Ricci flow in this context, but it's not as simple as other smoothing flows.

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  • $\begingroup$ Thanks! So if you don't mind me asking, what are some of those applications of Ricci flow to computer vision, etc? I'm now wondering why mean curvature flow wasn't used in those cases. $\endgroup$ – 900edges Mar 3 at 20:41
  • $\begingroup$ It is definitely not my area of expertise, but here is one paper that used Ricci flow for shape analysis. www3.cs.stonybrook.edu/~cvl/content/papers/2007/gu-iccv2007.pdf $\endgroup$ – Gabe K Mar 3 at 21:28

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