Sparse Bayesian Deep Functional Learning with Structured Region Selection

TL;DR

This paper introduces sBayFDNN, a Bayesian deep neural network that models complex functional data, offering interpretable region selection and strong theoretical guarantees, validated through simulations and real-world applications.

Contribution

It presents the first Bayesian deep functional model with theoretical guarantees and interpretable region selection for complex nonlinear data.

Findings

Effective in capturing nonlinear relationships

Accurately identifies influential regions

Outperforms existing methods in predictions

Abstract

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.