I have a few gaps in my understanding:

I know that PCA does dimensionality reduction. It does so by finding transformations such that the projection of the data points onto certain lines minimize the error of such projection (the Frobenius norm). PCA finds these lines, and these lines are linear combinations of existing variables.

Manifold hypothesis says that data must be intrinsically low dimensional.

My question is: since, PCA can only find linear transformations, so PCA can't do anything if the manifold is non-linear. Is any modification of PCA possible to extend it to find non linear manifolds?


Kernel PCA might be helpful. It performs linear PCA of the data projected in a reproducing kernel hilbert space where the data almost always lies on a linear manifold


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