7
$\begingroup$

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”

$\endgroup$
3
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.