How to define the dimensionality and manifoldness of point data?

Question

Consider the following scenario. A set of points is uniformly sampled from the surface of a sphere. Then a perturbation noise is applied to each point, displacing them in space.

As the magnitude of the random displacement increases the resulting spread of points makes it harder and harder to recognize the shape of the original sphere.

This same process can be done with a ball and similarly as the displacement increases the original shape becomes harder to discern.

My question is, given an arbitrary point cloud I want to give an estimate of the "manifoldness" and dimensionality of the underlying object.

So if I give you a sampling of the sphere with little displacement, then you tell me it's manifold and it's dimension is two (with some confidence). As the displacement increases the associated confidence to both manifoldness and dimensionality goes down.

Is there a particular resource I can use for this? Has this problem been explored?

if the manifold were a hyperplane, I'd say that an SVD would be a good approach — user619894, Oct 01 '25 at 21:20
Point cloud denoising, manifold learning, and nonlinear dimensionality reduction are all active areas of research. There is no well-established classical method for nonlinearly embedded manifolds. Check out the first section of this short survey and the included keywords and references — whpowell96, Oct 01 '25 at 21:26
One can use things like Vietoris-Rips complexes to reconstruct spaces (up to homotopy) given a finite sample of points near it, but maybe up to homotopy is not strong enough for what you want. — pyridoxal_trigeminus, Oct 02 '25 at 00:43

How to define the dimensionality and manifoldness of point data?

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