Title: Sparse Bayesian Deep Functional Learning with Structured Region Selection

Abstract:

Complex, continuously structured data are prevalent in contemporary fields such as industrial equipment diagnostics, wearable sensing, neuroimaging, and ECG monitoring. While this ubiquity offers new opportunities for functional data analysis, it also introduces significant challenges. Current methodologies are hindered by a fundamental dichotomy: traditional functional models are constrained by linear assumptions, while deep learning techniques generally fail to provide interpretable, sparse region selection.

To resolve these limitations, we introduce the sparse Bayesian functional deep neural network (sBayFDNN). This framework leverages a deep Bayesian architecture to learn adaptive functional embeddings, thereby capturing intricate nonlinear relationships. Simultaneously, it employs a structured prior to facilitate the interpretable, region-wise selection of influential domains, complete with quantified uncertainty.

From a theoretical standpoint, we derive rigorous bounds for approximation errors, alongside proofs of posterior consistency and consistency in region selection. These findings constitute the first theoretical guarantees for a Bayesian deep functional model, underscoring its statistical rigor and reliability. Empirical evaluations, encompassing both extensive simulations and real-world case studies, validate the superiority and effectiveness of sBayFDNN. Notably, the model outperforms existing approaches by more accurately identifying functionally significant regions and recognizing complex dependencies, leading to enhanced predictive performance.

Source: arXiv Generated at: 2026-06-04 00:00:00 UTC