# Functional Nonlinear Learning

arXiv: https://arxiv.org/abs/2206.11424 · HTML: https://mlsanity.com/paper/2206.11424/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 34 references that are in the map.
- First posted on arXiv 22 Jun 2022.

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

- [The UCR Time Series Archive](https://mlsanity.com/paper/1810.07758/) (2018)
- [Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains](https://mlsanity.com/paper/1509.02029/) (2015)
- [Optimal Bayes Classifiers for Functional Data and Density Ratios](https://mlsanity.com/paper/1605.03707/) (2016)
- [Localized Functional Principal Component Analysis](https://mlsanity.com/paper/1501.04933/) (2015)
- [Recovering the Underlying Trajectory from Sparse and Irregular Longitudinal Data](https://mlsanity.com/paper/1809.06462/) (2018)
- [Continuum centroid classifier for functional data](https://mlsanity.com/paper/2102.06130/) (2021)

## Built on by (most cited first, 3 of 3)

- [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)
- [Deep learning estimation of the spectral density of functional time series on large domains](https://mlsanity.com/paper/2601.00284/) (2026)

## Abstract (arXiv)

Using representations of functional data can be more convenient and beneficial in subsequent statistical models than direct observations. These representations, in a lower-dimensional space, extract and compress information from individual curves. The existing representation learning approaches in functional data analysis usually use linear mapping in parallel to those from multivariate analysis, e.g., functional principal component analysis (FPCA). However, functions, as infinite-dimensional objects, sometimes have nonlinear structures that cannot be uncovered by linear mapping. Linear methods will be more overwhelmed given multivariate functional data. For that matter, this paper proposes a functional nonlinear learning (FunNoL) method to sufficiently represent multivariate functional data in a lower-dimensional feature space. Furthermore, we merge a classification model for enriching the ability of representations in predicting curve labels. Hence, representations from FunNoL can be used for both curve reconstruction and classification. Additionally, we have endowed the proposed model with the ability to address the missing observation problem as well as to further denoise observations. The resulting representations are robust to observations that are locally disturbed by uncontrollable random noises. We apply the proposed FunNoL method to several real data sets and show that FunNoL can achieve better classifications than FPCA, especially in the multivariate functional data setting. Simulation studies have shown that FunNoL provides satisfactory curve classification and reconstruction regardless of data sparsity.