用多谐波级联建模复杂非线性函数，无需梯度下降即可快速训练。
Polyharmonic Cascade
- 通过随机函数理论构建多层多谐波样条，实现全局平滑逼近。
- 在MNIST上实现快速学习且不发生过拟合，训练效率高。
- 适合需要可解释性和平滑性的回归与生成任务研究者。
本文提出一种深度机器学习架构——多谐波级联（polyharmonic cascade），由一系列基于随机函数理论和无差别原理严格推导出的多谐波样条组成。该结构可逼近任意复杂度的非线性函数，同时保持全局光滑性和概率解释。针对该模型，提出一种替代梯度下降的训练方法：不直接优化系数，而是对每个批次求解一个关于固定节点“星群”处函数值的全局线性系统。该方法实现各层同步更新，保留单层的概率解释及理论一致性，并具备良好可扩展性：所有计算可归约为高效在GPU上执行的二维矩阵运算。在MNIST数据集上展示了快速学习且无过拟合的表现。
原文摘要 · Abstract (English)
This paper presents a deep machine learning architecture, the "polyharmonic cascade" -- a sequence of packages of polyharmonic splines, where each layer is rigorously derived from the theory of random functions and the principles of indifference. This makes it possible to approximate nonlinear functions of arbitrary complexity while preserving global smoothness and a probabilistic interpretation. For the polyharmonic cascade, a training method alternative to gradient descent is proposed: instead of directly optimizing the coefficients, one solves a single global linear system on each batch with respect to the function values at fixed "constellations" of nodes. This yields synchronized updates of all layers, preserves the probabilistic interpretation of individual layers and theoretical consistency with the original model, and scales well: all computations reduce to 2D matrix operations efficiently executed on a GPU. Fast learning without overfitting on MNIST is demonstrated.
Thank you to arXiv for use of its open access interoperability. PaperDance 不是 arXiv 官方产品；中文卡片由大模型生成，请以原文为准。