用分层多项调和样条解决高维回归，兼顾效率与理论正确性
Polyharmonic Spline Packages: Composition, Efficient Procedures for Computation and Differentiation
- 构建分层样条包架构，突破传统方法O(N³)计算瓶颈
- 在未知低维流形上仍保持理论有效性，支持高维数据建模
- 提供前向计算与端到端梯度的高效矩阵算法，适合大规模学习
此前研究证明，机器学习回归问题可在随机函数理论框架下求解，最优核函数由对称性与不变性原理导出，且等价于多项调和样条。然而，直接应用受限于O(N³)计算复杂度，且当输入空间维度过高时原理论假设失效。本文提出一种基于多项调和样条包的级联架构，同时解决可扩展性问题，并在输入具有未知低维流形结构时保持理论合理性。文中给出前向计算与端到端梯度传播的高效矩阵实现方法，适用于大规模学习任务。
原文摘要 · Abstract (English)
In a previous paper it was shown that a machine learning regression problem can be solved within the framework of random function theory, with the optimal kernel analytically derived from symmetry and indifference principles and coinciding with a polyharmonic spline. However, a direct application of that solution is limited by O(N^3) computational cost and by a breakdown of the original theoretical assumptions when the input space has excessive dimensionality. This paper proposes a cascade architecture built from packages of polyharmonic splines that simultaneously addresses scalability and is theoretically justified for problems with unknown intrinsic low dimensionality. Efficient matrix procedures are presented for forward computation and end-to-end differentiation through the cascade.
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