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I'm working on a project where we are embedding data into an n-dimensional Poincare ball similar to this paper. However, we'd like to take the additional step of reducing this data to a 2-dimensional hyperbolic manifold. Is there a way analogous to PCA that preserves the topological properties of the transformation in a somewhat similar way?

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  • $\begingroup$ It's not clear what you're asking about "topological properties," because topologically that ball is homeomorphic to the Euclidean space of the same dimension and PCA has fundamental geometric (not topological) meaning. One simple way to carry out PCA in hyperbolic space would be to perform the computations in the Beltrami-Klein model. Would that be the sort of thing you have in mind? $\endgroup$ – whuber Apr 26 '18 at 20:39

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