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Do every data resembles some form of space like manifold? Meaning, do geometry of data (especially the big data) is a manifold?

Is every data embedded to some sphere? If that is so, can we also say that whatever geometry of data after doing the dimensionality reduction is a (n-k) sphere, given that n is a dimension of the data before dimensionality reduction?

Basically, I am curious if geometry of every data would be a topological manifold (better yet, the differentiable manifold)...

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  • $\begingroup$ I think the term you’re looking for is “manifold learning”. $\endgroup$ – Dave May 19 at 2:26
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It is not the case that all data resembles some manifold, for most reasonable meanings of the phrase "resembles some manifold".

Mathematically, zero dimensional manifolds are collections of points, and technically speaking all finite data sets can be thought of as zero dimensional manifolds. However, I'm quite sure that's not what you had in mind when you asked the question.

There are many examples of data sets which do "resemble a shape", but for which the shape they resemble is not a topological manifold. A simple example is the space of points in the plane given by a letter X. This is a perfectly good space, but not a topological manifold. One can certainly produce data sets that "resemble that shape", and if they "resemble that shape" well enough, they probably don't resemble a topological manifold.

To address your third question, it is also not the case that after any process of dimensionality reduction data must lie on a (standard) sphere of some dimension, for most reasonable meanings of "lie on a sphere".

To really justify answers to any of your questions would require that we nail down precisely what we mean by these mathematical notions, but for the most intuitive meanings there are simple counterexamples to the statements appearing in your questions.

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