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vikigenius t1_iy7kxuu wrote

Reply to comment by olmec-akeru in [D] What method is state of the art dimensionality reduction by olmec-akeru

Huh? Diffeomorphisms are dimensionality preserving. You can't have a diffeomorphism between Rn to R2 unless n=2. That's the only way your differential mapping is bijective

So I am not sure how the diffeomorphisms guarantee that you can have lossless dimensionality reduction.

What can happen is that if your data inherently lies on a lower dimensional manifold. For instance if you have A subset of Rn that has an inherent dimensionality of just 2 then you can trivially just represent it in 2 dimensions. For example if you have a 3d space where the 3rd dimension is an exact linear combination of the 1st and 2nd then it's inherent dimensionality is 2 and you can obviously losslessly reduce it to 2d.

But most definitely not all datasets have an inherent dimensionality of 2.

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gooblywooblygoobly t1_iydf4z2 wrote

A super trivial example is that a (hyper)plane is a Riemannian manifold. Since we know that PCA is lossy, and PCA projects to a (hyper)plane, it can't be that projecting to manifolds are enough to perfectly preserve information.

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