Geometric data fitting

Given a dense set of points lying on or near an embedded submanifold M 0 ⊂ ℝ n of Euclidean space, the manifold fitting problem is to find an embedding F : M → ℝ n that approximates M 0 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 E d [ F ] , the expected square of the distance from a point in ℝ n to F ( M ) . It is shown that this approach to the fitting problem is guaranteed to fail because the functional E d 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.