# Sparse Bayesian Deep Functional Learning with Structured Region Selection

arXiv: https://arxiv.org/abs/2602.20651 · HTML: https://mlsanity.com/paper/2602.20651/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 50 references that are in the map.
- First posted on arXiv 24 Feb 2026.

## Builds on (most cited first, 16 of 16)

- [Optimal approximation of continuous functions by very deep ReLU networks](https://mlsanity.com/paper/1802.03620/) (2018)
- [LassoNet: A Neural Network with Feature Sparsity](https://mlsanity.com/paper/1907.12207/) (2019)
- [ECG Heartbeat Classification: A Deep Transferable Representation](https://mlsanity.com/paper/1805.00794/) (2018)
- [Consistent Feature Selection for Analytic Deep Neural Networks](https://mlsanity.com/paper/2010.08097/) (2020)
- [Consistent Sparse Deep Learning: Theory and Computation](https://mlsanity.com/paper/2102.13229/) (2021)
- [Deep Learning for Functional Data Analysis with Adaptive Basis Layers](https://mlsanity.com/paper/2106.10414/) (2021)
- [Deep Learning with Functional Inputs](https://mlsanity.com/paper/2006.09590/) (2020)
- [Supervised Feature Selection with Neuron Evolution in Sparse Neural Networks](https://mlsanity.com/paper/2303.07200/) (2023)
- [Functional Autoencoder for Smoothing and Representation Learning](https://mlsanity.com/paper/2401.09499/) (2024)
- [Bayesian Regularization for Graphical Models with Unequal Shrinkage](https://mlsanity.com/paper/1805.02257/) (2018)
- [Deep Neural Network Classifier for Multi-dimensional Functional Data](https://mlsanity.com/paper/2205.08592/) (2022)
- [Localized Functional Principal Component Analysis](https://mlsanity.com/paper/1501.04933/) (2015)
- [Functional Nonlinear Sparse Models](https://mlsanity.com/paper/1811.00577/) (2018)
- [Sparse-Input Neural Network using Group Concave Regularization](https://mlsanity.com/paper/2307.00344/) (2023)
- [Latent Multimodal Functional Graphical Model Estimation](https://mlsanity.com/paper/2210.17237/) (2022)
- [Smoothly Adaptively Centered Ridge Estimator](https://mlsanity.com/paper/2011.00289/) (2020)

## Abstract (arXiv)

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.