# Functional Autoencoders for Amplitude-Phase Representation Learning

arXiv: https://arxiv.org/abs/2609.34207 · HTML: https://mlsanity.com/paper/2609.34207/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 31 references that are in the map.
- First posted on arXiv 28 Sep 2026.

## Builds on (most cited first, 10 of 10)

- [Unsupervised Learning of Video Representations using LSTMs](https://mlsanity.com/paper/1502.04681/) (2015)
- [The UCR Time Series Archive](https://mlsanity.com/paper/1810.07758/) (2018)
- [LSTM-based Encoder-Decoder for Multi-sensor Anomaly Detection](https://mlsanity.com/paper/1607.00148/) (2016)
- [Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains](https://mlsanity.com/paper/1509.02029/) (2015)
- [Functional Data Analysis of Amplitude and Phase Variation](https://mlsanity.com/paper/1512.03216/) (2015)
- [Functional Autoencoder for Smoothing and Representation Learning](https://mlsanity.com/paper/2401.09499/) (2024)
- [Simultaneous Registration and Clustering for Multi-dimensional Functional Data](https://mlsanity.com/paper/1711.04761/) (2017)
- [Shape-Informed Clustering of Multi-Dimensional Functional Data via Deep Functional Autoencoders](https://mlsanity.com/paper/2509.22969/) (2025)
- [DeepFRC: An End-to-End Deep Learning Model for Functional Registration and Classification](https://mlsanity.com/paper/2501.18116/) (2025)
- [NeuralFLoC: Neural Flow-Based Joint Registration and Clustering of Functional Data](https://mlsanity.com/paper/2602.03169/) (2026)

## Abstract (arXiv)

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/.