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I did an exploration some time ago on using TDA tools to see how topological features change after application of some nonlinear dimensionality reduction methods.

For example I found out that, for Natural Image Patches dataset, reducing dimension using Isomap seems to preserve persistent homology better than LLE or Kernel PCA (authors of this paper talk about Klein bottle, which is 2d manifold, so I tried embedding in 3d space).

My question:

Is there any research on that?

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