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Sparse Bayesian Deep Functional Learning with Structured Region Selection

Xiaoxian Zhu, Yingmeng Li, Shuangge Ma, Mengyun Wu

TL;DR

Theoretically, these results provide the first theoretical guarantees for a Bayesian deep functional model, ensuring its reliability and statistical rigor and confirm the effectiveness and superiority of sBayFDNN.

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.

Sparse Bayesian Deep Functional Learning with Structured Region Selection

TL;DR

Theoretically, these results provide the first theoretical guarantees for a Bayesian deep functional model, ensuring its reliability and statistical rigor and confirm the effectiveness and superiority of sBayFDNN.

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.
Paper Structure (35 sections, 7 theorems, 101 equations, 8 figures, 2 tables, 1 algorithm)

This paper contains 35 sections, 7 theorems, 101 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Classical functional regression methods
  4. Deep Learning for Functional Data
  5. Sparsity and Selection in Neural Networks
  6. Methodology
  7. Optimization-based Bayesian Inference
  8. MAP fitting
  9. MAP plug-in posterior inclusion probabilities
  10. Mapping selected features to an active region on $\mathcal{T}$
  11. Theoretical Properties
  12. Experiments
  13. Simulation Studies
  14. Real data analysis
  15. Conclusion
  16. ...and 20 more sections

Key Result

Theorem 5.4

Suppose that Assumptions assumption_design_boundedness--assumption_Holder_link hold. Then:

Figures (8)

  • Figure 1: Workflow of sBayFDNN. The functional predictor $X_i(t)$ is projected onto locally supported B-spline bases to form spline features $\boldsymbol{x}_i=(x_{i1},\ldots,x_{i,J_n})^{\top}$. A DNN with a spike-and-slab prior on the first-layer weight columns yields feature-wise posterior inclusion probabilities (PIPs) $q_j$, which are thresholded to select spline features and mapped back to the function domain to produce an estimated active region $\widehat{\Omega}$.
  • Figure 2: F1 scores across $g$ functions, SNR settings, and $\beta(t)$ scenarios.
  • Figure 3: PIPs from sBayFDNN in the high-SNR, Medium-$\beta$, logistic-link scenario.
  • Figure 4: RMSE across $g$ functions, SNR settings, and $\beta(t)$ scenarios.
  • Figure 5: PIPs from sBayFDNN for ECG and Tecator datasets.
  • ...and 3 more figures

Theorems & Definitions (11)

  • Theorem 5.4
  • Theorem 5.7
  • Theorem 5.9
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • ...and 1 more