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!
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$\begingroup$ 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 $\endgroup$– Alberto SinigagliaAug 1, 2022 at 17:17
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$\begingroup$ @AlbertoSinigaglia The phrasing "preserves distances" is a common synonym for "isometry". I am not aware of any other meaning. $\endgroup$– GalenAug 1, 2022 at 17:22
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$\begingroup$ @Galen I'm referring to this (eg) stats.stackexchange.com/questions/176672/… $\endgroup$– Alberto SinigagliaAug 1, 2022 at 17:24
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$\begingroup$ also, non linear mapping usually has not closed form solution, and thus you can non linear autoencoders (idk about this isometry property though) $\endgroup$– Alberto SinigagliaAug 1, 2022 at 17:26
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1$\begingroup$ @AlbertoSinigaglia Interesting. Sort of a loose phrasing of "preserving", but I can imagine there being an asymptotic result applying to this. $\endgroup$– GalenAug 1, 2022 at 17:28
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