Sparse Bayesian Deep Functional Learning with Structured Region Selection

2026-02-24

AI Summary

This work proposes a sparse Bayesian functional deep neural network (sBayFDNN) to reconcile the limited expressiveness of traditional functional linear models with the lack of interpretable, sparse region selection in deep learning. By integrating deep neural networks with structured sparse Bayesian priors, sBayFDNN adaptively learns functional embeddings to capture complex nonlinear relationships while enabling interpretable region selection with quantified uncertainty. Theoretically, this paper establishes the first approximation error bounds, posterior consistency, and region selection consistency guarantees for Bayesian deep functional models. Empirically, the method demonstrates superior performance over existing approaches on both simulated and real-world data—including ECG and neuroimaging datasets—achieving high predictive accuracy alongside reliable identification of relevant functional regions.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

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.

Problem

• functional data analysis
• sparse effects
• interpretable region selection
• nonlinear relationships
• structured data

Innovation

• Sparse Bayesian
• Functional Data Analysis
• Deep Neural Networks
• Structured Region Selection
• Posterior Consistency