Hessian Based Smoothing Splines for Manifold Learning
This provides a theoretically grounded smoothing spline method for manifold learning applications, though it appears incremental as it builds on existing Hessian estimation techniques.
The authors tackled the problem of extending smoothing splines to manifold learning by generalizing the bending energy penalty to manifolds using the Hessian's Frobenius norm, resulting in a method that minimizes square error with a global curvature penalty and provides existence/uniqueness guarantees.
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.