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My (not too deep) understanding of the curse of dimensionality that affects a classification algorithm, such as k-Nearest Neighbor, is that at higher dimensions the 'sparsity' of euclidean space kicks in (this can be seen, for example by comparing the volume/content/measure of the unit ball with respect to the unit box)

I wanted to know if researchers have considered working on riemannian manifolds other than euclidean space (or spaces with other $L_p$ norms ($p \neq 2$)$^1$), say with metric tensor $\sum g_{ij} dx^i \otimes dx^j$ where the coefficients $g_{ij}$ are non-constant or maybe even dependent on the data?

1.Aggarwal C. C., Hinneburg A., Keim, D. A. (2001), ”On the Surprising Behavior of Distance Metrics in High Dimensional Space”

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Yes indeed, that is the case of the Isomap algorithm and Laplacians eigenmaps, for example. There are quite a few approaches listed in this article and also here. Some of them are implemented in scikit-learn.

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