# Deep Learning for Functional Data Analysis with Adaptive Basis Layers

arXiv: https://arxiv.org/abs/2106.10414 · HTML: https://mlsanity.com/paper/2106.10414/

## Key facts

- Cited by 17 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 41 references that are in the map.
- First posted on arXiv 19 Jun 2021.

## Builds on (most cited first, 6 of 6)

- [Deep Learning and the Information Bottleneck Principle](https://mlsanity.com/paper/1503.02406/) (2015)
- [Network In Network](https://mlsanity.com/paper/1312.4400/) (2013)
- [Train faster, generalize better: Stability of stochastic gradient descent](https://mlsanity.com/paper/1509.01240/) (2015)
- [Sampling for Inference in Probabilistic Models with Fast Bayesian Quadrature](https://mlsanity.com/paper/1411.0439/) (2014)
- [On Universal Approximation by Neural Networks with Uniform Guarantees on Approximation of Infinite Dimensional Maps](https://mlsanity.com/paper/1910.01545/) (2019)
- [Deep Function Machines: Generalized Neural Networks for Topological Layer Expression](https://mlsanity.com/paper/1612.04799/) (2016)

## Built on by (most cited first, 17 of 17)

- [MM-DAG: Multi-task DAG Learning for Multi-modal Data -- with Application for Traffic Congestion Analysis](https://mlsanity.com/paper/2306.02831/) (2023)
- [Nonlinear functional regression by functional deep neural network with kernel embedding](https://mlsanity.com/paper/2401.02890/) (2024)
- [Functional Autoencoder for Smoothing and Representation Learning](https://mlsanity.com/paper/2401.09499/) (2024)
- [Neural Networks for Scalar Input and Functional Output](https://mlsanity.com/paper/2208.05776/) (2022)
- [Shape-Informed Clustering of Multi-Dimensional Functional Data via Deep Functional Autoencoders](https://mlsanity.com/paper/2509.22969/) (2025)
- [Practical Performative Policy Learning with Strategic Agents](https://mlsanity.com/paper/2412.01344/) (2024)
- [DeepFRC: An End-to-End Deep Learning Model for Functional Registration and Classification](https://mlsanity.com/paper/2501.18116/) (2025)
- [On regularized polynomial functional regression](https://mlsanity.com/paper/2311.03036/) (2023)
- [Functional relevance based on the continuous Shapley value](https://mlsanity.com/paper/2411.18575/) (2024)
- [A Doubly Robust Machine Learning Approach for Disentangling Treatment Effect Heterogeneity with Functional Outcomes](https://mlsanity.com/paper/2602.11118/) (2026)
- [Sparse-Aware Neural Networks for Nonlinear Functionals: Mitigating the Exponential Dependence on Dimension](https://mlsanity.com/paper/2604.06774/) (2026)
- [Deep Functional Factor Models: Forecasting High-Dimensional Functional Time Series via Bayesian Nonparametric Factorization](https://mlsanity.com/paper/2305.14543/) (2023)
- [Deep Survival Analysis for Competing Risk Modeling with Functional Covariates and Missing Data Imputation](https://mlsanity.com/paper/2509.25381/) (2025)
- [Deep learning estimation of the spectral density of functional time series on large domains](https://mlsanity.com/paper/2601.00284/) (2026)
- [Sparse Bayesian Deep Functional Learning with Structured Region Selection](https://mlsanity.com/paper/2602.20651/) (2026)
- [Deep Shape Regression for Planar Curves with Multimodal Covariates](https://mlsanity.com/paper/2607.19600/) (2026)
- [Approximating Smooth Functionals with ReLU Networks](https://mlsanity.com/paper/2609.15355/) (2026)

## Abstract (arXiv)

Despite their widespread success, the application of deep neural networks to functional data remains scarce today. The infinite dimensionality of functional data means standard learning algorithms can be applied only after appropriate dimension reduction, typically achieved via basis expansions. Currently, these bases are chosen a priori without the information for the task at hand and thus may not be effective for the designated task. We instead propose to adaptively learn these bases in an end-to-end fashion. We introduce neural networks that employ a new Basis Layer whose hidden units are each basis functions themselves implemented as a micro neural network. Our architecture learns to apply parsimonious dimension reduction to functional inputs that focuses only on information relevant to the target rather than irrelevant variation in the input function. Across numerous classification/regression tasks with functional data, our method empirically outperforms other types of neural networks, and we prove that our approach is statistically consistent with low generalization error. Code is available at: urlhttps://github.com/jwyyy/AdaFNN.