Geometric data fitting

Abstract

Given a dense set of points lying on or near an embedded submanifold $M_0\subset {ℝ}^n$ of Euclidean space, the manifold fitting problem is to find an embedding $F:M\to {ℝ}^n$ 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 $E_d\left[F\right]$, the expected square of the distance from a point in ${ℝ}^n$ to $F\left(M\right)$. 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.