This paper addresses the dimensionality reduction of multivariable function data, where observations are presented as labeled correlated curves on a shared time network, typically sampled on discrete and fuzzy networks. Given the complexity of the interactions between variables and time, we train a small latent space of functions using a functional autoencoder (FAE) to handle novel and unprecedented multivariable smooth trajectories. The experiment assumes a tightly controlled simulation design to test the method under realistic sampling conditions, where the sample size (n), time network resolution (Time), number of Fourier basis functions (φ), and noise level are varied. Several complementary performance metrics are used to determine the quality of dimensionality reduction: the mean integrated error (IMSE), variance conservation ratio (VPR), and Spearman's correlation coefficient (ρ) to determine the overall geometry, and T(k) reliability to determine the local geometry of the reduced space. The results illustrate the trade-off between reconstructing and preserving the structure under different conditions, providing useful information about when functional autoencoders can be relied upon to be useful in multivariable functional analysis, preserving local structure, and reducing dimensions.
Keywords
Functional Data AnalysisMultivariate Functional DataFunctional autoencoderRepresentation learningDimensionality Reduction.
References
S. Singh, S. Coyle, and M. Zhang, “Shape-Informed Clustering of Multi-Dimensional Functional Data via Deep Functional Autoencoders,” in 39th Conference on Neural Information Processing Systems (NeurIPS 2025), Vancouver, BC, Canada, Dec. 2025, [Online]. Available: https://openreview.net/forum?id=5mpQO0YpTv.
S. Golovkine, E. Gunning, A. J. Simpkin, and N. Bargary, “On the estimation of the number of components in multivariate functional principal component analysis,” Communications in Statistics - Simulation and Computation, 2025, [Online]. Available: https://doi.org/10.1080/03610918.2025.2459862.
Y. Zhou et al., “Multimodal functional deep learning for multiomics data,” Briefings in Bioinformatics, vol. 25, no. 5, Sep. 2024, [Online]. Available: https://doi.org/10.1093/bib/bbae448.
G. Aneiros and P. Vieu, “Variable selection in infinite-dimensional problems,” Statistics & Probability Letters, vol. 94, pp. 12-20, 2014, [Online]. Available: https://doi.org/10.1016/j.spl.2014.06.025.
Y. Li, Y. Qiu, and Y. Xu, “From multivariate to functional data analysis: Fundamentals, recent developments, and emerging areas,” Journal of Multivariate Analysis, vol. 188, pp. 1-21, 2022, [Online]. Available: https://doi.org/10.1016/j.jmva.2021.104806.
T.-Y. Hsieh, Y. Sun, S. Wang, and V. Honavar, “Functional Autoencoders for Functional Data Representation Learning,” 2021, [Online]. Available: https://epubs.siam.org/terms-privacy.
S. Wu, C. Beaulac, and J. Cao, “Functional autoencoder for smoothing and representation learning,” Statistical Computing, vol. 34, p. 203, 2024, [Online]. Available: https://doi.org/10.1007/s11222-024-10501-w.
J. Yao and J. Mueller, “Deep Learning for Functional Data Analysis with Adaptive Basis Layers,” 2021, [Online]. Available: https://doi.org/10.48550/arXiv.2106.10414.
G. Aneiros, R. Cao, R. Fraiman, C. Genest, and P. Vieu, “Recent advances in functional data analysis and high-dimensional statistics,” Journal of Multivariate Analysis, vol. 170, pp. 3-9, 2019, [Online]. Available: https://doi.org/10.1016/j.jmva.2018.11.007.
F. Ieva and A. M. Paganoni, “Depth measures for multivariate functional data,” in Communications in Statistics - Theory and Methods, 2013, pp. 1265-1276, [Online]. Available: https://doi.org/10.1080/03610926.2012.746368.
P. Laforgue and S. Clémençon, “Autoencoding any Data through Kernel Autoencoders,” 2019.
P. Bickel, P. Diggle, S. Fienberg, U. Gather, I. Olkin, and S. Zeger, Springer Series in Statistics, doi: 10.1007/978-3-642-20192-9.
