将数学原理与深度学习结合，实现高精度可解释的统计保证模型。
Wahkon: A Statistically Principled Deep RKHS Superposition Network
- 基于核空间正则化与叠加定理构建深层网络，支持逐层复杂度控制。
- 理论证明其估计为最大后验概率解，收敛速度达到最优。
- 在基因数据等场景表现优于传统神经网络，适合追求可信预测的领域。
深度学习擅长预测但常缺乏有限样本保证和校准的不确定性；基于再生核希尔伯特空间（RKHS）的方法虽能提供这些保证，却难以适应高维数据。本文提出Wahkon，一种将柯尔莫戈洛夫叠加定理与RKHS正则化结合的深层超叠加网络。该方法基于瓦尔巴的平滑样条传统，导出有限维深度表示定理，使训练可行并实现显式的逐层复杂度控制。我们证明了惩罚估计器正是在分层高斯过程先验下的最大后验估计，将样条与高斯过程对偶性扩展至深层复合结构。通过度量熵分析，我们在温和光滑条件下建立了最小最大最优收敛速率，并阐明深度与宽度如何与正则性权衡。实验表明，Wahkon在模拟基准和单细胞CITE-seq研究中均优于多层感知机、神经正切核及柯尔莫戈洛夫-阿诺德网络。通过统一柯尔莫戈洛夫叠加定理与RKHS正则化，Wahkon在一个框架内实现了准确性、可解释性与统计严谨性的统一。
原文摘要 · Abstract (English)
Deep learning excels at prediction but often lacks finite-sample guarantees and calibrated uncertainty; RKHS (Reproducing Kernel Hilbert Space)-based methods provide those guarantees but struggle to adapt in high dimensions. We propose Wahkon, a deep RKHS superposition network that unifies Kolmogorov's superposition principle with RKHS regularization in the smoothing-spline tradition of Wahba. This yields a finite-dimensional deep representer theorem that makes training tractable and provides explicit layerwise complexity control. We show the penalized estimator is exactly the MAP (maximum a posteriori) estimate under a hierarchical Gaussian-process prior, extending the spline/GP duality to deep compositions. Using metric-entropy arguments, we establish minimax-optimal convergence rates under mild smoothness and clarify how depth and width trade off with regularity. Empirically, Wahkon outperforms multilayer perceptrons, Neural Tangent Kernels, and Kolmogorov--Arnold Networks across simulation benchmarks and a single-cell CITE-seq study. By unifying Kolmogorov's superposition principle with RKHS regularization, Wahkon delivers accuracy, interpretability, and statistical rigor in a single framework.
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