  We propose a multidimensional smoothing spline algorithm in the context of
manifold learning. We generalize the bending energy penalty of thin-plate
splines to a quadratic form on the Sobolev space of a flat manifold, based on
the Frobenius norm of the Hessian matrix. This leads to a natural definition of
smoothing splines on manifolds, which minimizes square error while optimizing a
global curvature penalty. The existence and uniqueness of the solution is shown
by applying the theory of reproducing kernel Hilbert spaces. The minimizer is
expressed as a combination of Green's functions for the biharmonic operator,
and 'linear' functions of everywhere vanishing Hessian. Furthermore, we utilize
the Hessian estimation procedure from the Hessian Eigenmaps algorithm to
approximate the spline loss when the true manifold is unknown. This yields a
particularly simple quadratic optimization algorithm for smoothing response
values without needing to fit the underlying manifold. Analysis of asymptotic
error and robustness are given, as well as discussion of out-of-sample
prediction methods and applications.
