Geometric data fitting (Q1773457)

The author solves the following problem: \(M\) be a smooth, compact manifold without boundary; \(M_0\) be the image of a smooth embedding \(F_0:M\to \mathbb R^n\). Let \(Y=\{y_1,-,y_n\}\subset \mathbb R^n\) be a collection of points contained in a smooth tubular neighborhood \(\Omega\) of \(M_0\). The manifold fitting problem is to find an embedding \(F:M\to \mathbb R^n\) such that \(F(M)\) is a good approximation to \(M_0\) in the sense of least squares. It is shown that this approach to the fitting problem is guaranteed to fail because the functional of expected square of the distance from a point in \(\mathbb R^n\) to \(F(M)\) has no local minima. This problem is addressed by adding a small multiple \(k\) of the energy functional to the expected square of the distance. The calculus of variations is used for the study of the problem.