Abstract

Given a dense set of points lying on or near an embedded submanifold M0⊂ℝn of Euclidean space, the manifold fitting problem is to find an embedding F:M→ℝn that approximates M0 in the sense of least squares. When the dataset is modeled by a probability distribution, the fitting problem reduces to that of finding an embedding that minimizes Ed[F], the expected square of the distance from a point in ℝn to F(M). It is shown that this approach to the fitting problem is guaranteed to fail because the functional Ed has no local minima. This problem is addressed by adding a small multiple k of the harmonic energy functional to the expected square of the distance. Techniques from the calculus of variations are then used to study this modified functional.