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：我们提出了一种适用于流形学习场景的多维平滑样条算法。通过基于Hessian矩阵的Frobenius范数，将薄板样条的弯曲能量惩罚项推广至平坦流形Sobolev空间中的二次型形式。这自然定义了流形上的平滑样条——在最小化平方误差的同时优化全局曲率惩罚项。利用再生核希尔伯特空间理论证明了该解的存在唯一性，其极小元可表示为双调和算子的格林函数与Hessian处处为零的"线性"函数的线性组合。进一步地，当真实流形未知时，我们采用Hessian特征映射算法中的Hessian估计方法近似样条损失函数。该方法无需拟合底层流形即可实现极为简洁的二次优化算法来平滑响应值。本文给出了渐近误差分析与鲁棒性证明，并讨论了样本外预测方法和应用场景。