  We develop theory for nonlinear dimensionality reduction (NLDR). A number of
NLDR methods have been developed, but there is limited understanding of how
these methods work and the relationships between them. There is limited basis
for using existing NLDR theory for deriving new algorithms. We provide a novel
framework for analysis of NLDR via a connection to the statistical theory of
linear smoothers. This allows us to both understand existing methods and derive
new ones. We use this connection to smoothing to show that asymptotically,
existing NLDR methods correspond to discrete approximations of the solutions of
sets of differential equations given a boundary condition. In particular, we
can characterize many existing methods in terms of just three limiting
differential operators and boundary conditions. Our theory also provides a way
to assert that one method is preferable to another; indeed, we show Local
Tangent Space Alignment is superior within a class of methods that assume a
global coordinate chart defines an isometric embedding of the manifold.
