Given a dense set of points lying on or near an embedded
submanifold $M_0\subset {\mathbb R}^n$ of Euclidean
space, the manifold fitting problem is to find an
embedding $F : M \rightarrow {\mathbb R}^n$ 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 ${\mathbb R}^n$
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.