Abstract and Applied Analysis
Volume 2004 (2004), Issue 10, Pages 831-880

Geometric data fitting

José L. Martínez-Morales

Instituto de Matemáticas, Universidad Nacional Autónoma de México, A.P. 273, Admon. de Correos #3, Cuernavaca C.P. 62251, Morelos, Mexico

Received 21 April 2003

Copyright © 2004 José L. Martínez-Morales. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Given a dense set of points lying on or near an embedded submanifold M0n of Euclidean space, the manifold fitting problem is to find an embedding F:Mn 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 n 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.