Suppose that a space ${\mathbb R}^{r}$ contains a set of points which we want to consider as enclosing a volume within the space, or perhaps a volume in a submanifold (e.g., the sphere $S^{2}$ within 3-space). How do we detect computationally that a set of points constitutes a manifold? Some cases such as regularly spaced rectangular arrays are easy, but I will be looking at irregularly distributed points, and they won't necessarily look "nice". Any literature on this subject? Thanks.
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Are you taking a random sampling of points in a known submanifold and looking for ways to find out what that submanifold is? – Dan Rust May 28 '13 at 00:57
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Strictly speaking, any discrete set of points is contained in some submanifold. In practice one can look for submanifolds that are reasonably simple: defined by equation $F(x,y,z)=0$ where $F$ is a function in some class. A key term here is surface fitting. – ˈjuː.zɚ79365 May 28 '13 at 01:54
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You may also be interested in random manifolds. – Dan Rust May 28 '13 at 02:25
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See, for example, http://en.wikipedia.org/wiki/Nonlinear_dimensionality_reduction#Manifold_learning_algorithms . – Qiaochu Yuan May 28 '13 at 02:59