Which assumption posits that high-dimensional data lies on lower-dimensional substructures called topological manifolds?

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The manifold assumption explains this idea: high-dimensional data actually lies on or near a lower-dimensional, curved surface called a manifold embedded in the ambient space. This means the true degrees of freedom are few, even though data points have many coordinates, and nonlinear methods often reveal that underlying structure by mapping or unfolding the manifold.

Topological manifolds describe the shape of these surfaces—spaces that locally look like Euclidean space but can be globally curved. The term refers to the object, not the claim about the data. The manifold assumption is the stated idea about data geometry.

Why the other terms don’t fit: a topological manifold is the mathematical object itself, not the assumption about data; high-dimensional data describes the data, not its geometric claim; PCA is a linear technique that assumes data lie near a linear subspace, which isn’t adequate for curved manifolds.