# Hessian Based Smoothing Splines for Manifold Learning

arXiv: https://arxiv.org/abs/2302.05025 · HTML: https://mlsanity.com/paper/2302.05025/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 1 of its 34 references that are in the map.
- First posted on arXiv 10 Feb 2023.

## Builds on (most cited first, 1 of 1)

- [Smoothing splines on Riemannian manifolds, with applications to 3D shape space](https://mlsanity.com/paper/1801.04978/) (2018)

## Abstract (arXiv)

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.