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.

翻译：我们建议了多元学习背景下的多维平滑样条算法。 我们根据赫森矩阵的Frobenius规范,将薄板样条的弯曲能量罚款推广到Sobolev空间的平面体形体形。 这导致对在多元体上平滑样条的自然定义, 从而在优化全球曲线惩罚的同时将平滑样条错误最小化。 解决方案的存在和独特性通过应用再生产内核Hilbert空间的理论来显示。 最小化器表现为绿色功能的组合, 用于双声管操作器, 以及“ 线性” 功能的组合。 此外, 我们使用赫森 Eigenmaps 的赫斯估计程序, 以在未知真实的元体外测算方法和应用时, 来估计浮点损失。 这产生一种特别简单的二次优化算法, 以平滑动反应值而无需适应根本的多元性。 分析虚伪的错误和稳健性, 以及讨论悬浮性预测方法和应用。