arXiv:2602.20651cs.LGstat.AP2026-02

提出可解释的稀疏贝叶斯深度函数模型,精准定位重要数据区域。

Sparse Bayesian Deep Functional Learning with Structured Region Selection

  • 基于结构化先验的贝叶斯深度网络,实现非线性关系建模与可解释区域选择。
  • 理论证明逼近误差、后验一致性和区域选择一致性,首次提供贝叶斯深度函数模型的理论保障。
  • 在心电图、神经影像等真实数据中表现更优,适合需要可解释性的高精度预测场景。

在心电图监测、脑成像、可穿戴传感和工业设备诊断等现代应用中,复杂且连续的结构化数据普遍存在,为函数数据分析带来挑战与机遇。然而,现有方法面临关键权衡:传统函数模型受限于线性假设,而深度学习方法缺乏对稀疏效应的可解释区域选择能力。为此,我们提出稀疏贝叶斯函数深度神经网络(sBayFDNN)。该模型通过深度贝叶斯架构学习自适应函数嵌入,捕捉复杂的非线性关系;同时,利用结构化先验实现可解释的、逐域的影响区域选择,并量化不确定性。理论上,我们建立了严格的逼近误差界、后验一致性和区域选择一致性结果,首次为贝叶斯深度函数模型提供了理论保证,确保其可靠性与统计严谨性。实验上,通过全面模拟与真实世界研究验证了sBayFDNN的有效性与优越性。关键在于,sBayFDNN能准确识别复杂依赖关系并更精确地定位功能有意义的区域,这一能力远超现有方法。

原文摘要 · Abstract (English)

In modern applications such as ECG monitoring, neuroimaging, wearable sensing, and industrial equipment diagnostics, complex and continuously structured data are ubiquitous, presenting both challenges and opportunities for functional data analysis. However, existing methods face a critical trade-off: conventional functional models are limited by linearity, whereas deep learning approaches lack interpretable region selection for sparse effects. To bridge these gaps, we propose a sparse Bayesian functional deep neural network (sBayFDNN). It learns adaptive functional embeddings through a deep Bayesian architecture to capture complex nonlinear relationships, while a structured prior enables interpretable, region-wise selection of influential domains with quantified uncertainty. Theoretically, we establish rigorous approximation error bounds, posterior consistency, and region selection consistency. These results provide the first theoretical guarantees for a Bayesian deep functional model, ensuring its reliability and statistical rigor. Empirically, comprehensive simulations and real-world studies confirm the effectiveness and superiority of sBayFDNN. Crucially, sBayFDNN excels in recognizing intricate dependencies for accurate predictions and more precisely identifies functionally meaningful regions, capabilities fundamentally beyond existing approaches.

函数数据分析贝叶斯深度学习可解释性稀疏建模

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