Sparse Bayesian Deep Functional Learning with Structured Region Selection
This addresses the problem of interpretable region selection in functional data analysis for applications like ECG monitoring and neuroimaging, representing a novel method for a known bottleneck.
The paper tackles the challenge of analyzing complex functional data where existing methods trade off between linearity and interpretability, proposing a sparse Bayesian functional deep neural network (sBayFDNN) that achieves accurate predictions and precisely identifies functionally meaningful regions with theoretical guarantees.
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.