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The question is the same as posed in the title; is there an alternative to PCA that doesn't rely on the linear assumption but maintains distances (i.e. the main issue with UMAP/tSNE)?

Thanks!

Yousuf Khan
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  • what you mean with "preserves distances"? that two points far 2 units one domain, should be far 2 in the second domain?... AFAIK, PCA preserves only "far" distances – Alberto Aug 01 '22 at 17:17
  • @AlbertoSinigaglia The phrasing "preserves distances" is a common synonym for "isometry". I am not aware of any other meaning. – Galen Aug 01 '22 at 17:22
  • @Galen I'm referring to this (eg) https://stats.stackexchange.com/questions/176672/what-is-meant-by-pca-preserving-only-large-pairwise-distances – Alberto Aug 01 '22 at 17:24
  • also, non linear mapping usually has not closed form solution, and thus you can non linear autoencoders (idk about this isometry property though) – Alberto Aug 01 '22 at 17:26
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    @AlbertoSinigaglia Interesting. Sort of a loose phrasing of "preserving", but I can imagine there being an asymptotic result applying to this. – Galen Aug 01 '22 at 17:28
  • Maybe a better way to phrase it is that I would like to know which features contribute to the differences in classes more than the non-important ones... like a ranking. For example in PCA I can check the variance contributed by each feature – Yousuf Khan Aug 02 '22 at 19:50

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