Functional Autoencoders for Amplitude-Phase Representation Learning
Abstract
Functional data are intrinsically infinite-dimensional, and often exhibit phase variation, where corresponding events occur at different times across observations. Existing linear dimension reduction methods struggle with nonlinear amplitude variation, while functional autoencoders without an explicit warp entangle temporal misalignment with shape. We propose the Amplitude--Phase Functional Autoencoders (AP-FAE), an unsupervised framework for functional data that spans both univariate and multivariate cases, with emphasis on the multivariate setting, and factorizes the latent space into separate amplitude and phase embeddings derived from all channels. A smooth functional decoder reconstructs channel-specific amplitude functions in canonical time, and a shared monotone, endpoint-preserving warp captures phase variation. We prove a bound linking amplitude recovery to registration, reconstruction, and noise errors, and validate it numerically. Across synthetic data and six real-world benchmarks, AP-FAE outperforms state-of-the-art baselines on most clustering and alignment metrics and on all reconstruction metrics. Clustering with amplitude embeddings alone consistently surpasses joint amplitude--phase clustering, confirming the benefit of explicit disentanglement. Code is available at https://anonymous.4open.science/r/APFAE-418C/}{https://anonymous.4open.science/r/APFAE-418C/.
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BibTeX
@misc{wu2026functional,
title = {{Functional Autoencoders for Amplitude-Phase Representation Learning}},
author = {Peida Wu and Xinyang Xiong and Pengcheng Zeng},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.34207},
eprint = {2609.34207},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.34207},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Functional Autoencoders for Amplitude-Phase Representation Learning
AU - Peida Wu
AU - Xinyang Xiong
AU - Pengcheng Zeng
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.34207
UR - https://arxiv.org/abs/2609.34207
AB - Functional data are intrinsically infinite-dimensional, and often exhibit phase variation, where corresponding events occur at different times across observations. Existing linear dimension reduction methods struggle with nonlinear amplitude variation, while functional autoencoders without an explicit warp entangle temporal misalignment with shape. We propose the Amplitude--Phase Functional Autoencoders (AP-FAE), an unsupervised framework for functional data that spans both univariate and multivariate cases, with emphasis on the multivariate setting, and factorizes the latent space into separate amplitude and phase embeddings derived from all channels. A smooth functional decoder reconstructs channel-specific amplitude functions in canonical time, and a shared monotone, endpoint-preserving warp captures phase variation. We prove a bound linking amplitude recovery to registration, reconstruction, and noise errors, and validate it numerically. Across synthetic data and six real-world benchmarks, AP-FAE outperforms state-of-the-art baselines on most clustering and alignment metrics and on all reconstruction metrics. Clustering with amplitude embeddings alone consistently surpasses joint amplitude--phase clustering, confirming the benefit of explicit disentanglement. Code is available at https://anonymous.4open.science/r/APFAE-418C/}{https://anonymous.4open.science/r/APFAE-418C/.
ER -