Abstract
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). In this article it is shown that this approach to the fitting problem is guaranteed to fail because the functional E d has no local minima. This explains why cross-validation does not appear to be a viable method for choosing the complexity of principal manifold estimates.
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