用哈尔基函数提升高维函数逼近精度，减少调参。
KAN/H: Kolmogorov-Arnold Network using Haar-like bases
- 用哈尔类分层基替代B样条，增强局部拟合能力。
- 在高维函数逼近和MNIST上表现优异，超参数少。
- 支持无界实值输入，适合机器学习中的复杂拟合任务。
使用哈尔基系统进行函数逼近，在通过帕特里夏树压缩时具有高效实现，同时保留了小波在全局与局部拟合中的灵活性。然而，如同基于B样条的逼近方法，高维下实现高精度仍具挑战。本文提出KAN/H，一种基于哈尔类分层基系统的柯尔莫戈罗夫-阿诺德网络（KAN）变体，其基函数具有非零一阶导数，替代原有B样条。我们还提出了学习率调度方法和处理无界实值输入的方法，利用线性逼近中哈尔类分层基的性质。将该算法应用于函数逼近问题及MNIST数据集，结果表明该方法所需的问题特定超参数极少。
原文摘要 · Abstract (English)
Function approximation using Haar basis systems offers an efficient implementation when compressed via Patricia trees while retaining the flexibility of wavelets for both global and local fitting. However, like B-spline-based approximations, achieving high accuracy in high dimensions remains challenging. This paper proposes KAN/H, a variant of the Kolmogorov-Arnold Network (KAN) that uses a Haar-like hierarchical basis system with nonzero first-order derivatives, instead of B-splines. We also propose a learning-rate scheduling method and a method for handling unbounded real-valued inputs, leveraging properties of linear approximation with Haar-like hierarchical bases. By applying the resulting algorithm to function-approximation problems and MNIST, we confirm that our approach requires minimal problem-specific hyperparameter tuning.
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