# Shape-Informed Clustering of Multi-Dimensional Functional Data via Deep Functional Autoencoders

arXiv: https://arxiv.org/abs/2509.22969 · HTML: https://mlsanity.com/paper/2509.22969/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 40 references that are in the map.
- First posted on arXiv 26 Sep 2025.

## Builds on (most cited first, 14 of 14)

- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Poincar'e Embeddings for Learning Hierarchical Representations](https://mlsanity.com/paper/1705.08039/) (2017)
- [Multiwavelet-based Operator Learning for Differential Equations](https://mlsanity.com/paper/2109.13459/) (2021)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [Universal Approximation Theorems for Differentiable Geometric Deep Learning](https://mlsanity.com/paper/2101.05390/) (2021)
- [Non-Euclidean Universal Approximation](https://mlsanity.com/paper/2006.02341/) (2020)
- [Variational Autoencoding Neural Operators](https://mlsanity.com/paper/2302.10351/) (2023)
- [Deep Learning for Functional Data Analysis with Adaptive Basis Layers](https://mlsanity.com/paper/2106.10414/) (2021)
- [Deep Learning with Functional Inputs](https://mlsanity.com/paper/2006.09590/) (2020)
- [Functional Neural Networks: Shift invariant models for functional data with applications to EEG classification](https://mlsanity.com/paper/2301.05869/) (2023)
- [On Universal Approximation by Neural Networks with Uniform Guarantees on Approximation of Infinite Dimensional Maps](https://mlsanity.com/paper/1910.01545/) (2019)
- [A Non-linear Function-on-Function Model for Regression with Time Series Data](https://mlsanity.com/paper/2011.12378/) (2020)
- [Review of Clustering Methods for Functional Data](https://mlsanity.com/paper/2210.00847/) (2022)

## Built on by (most cited first, 4 of 4)

- [Hypernetwork-based approach for grid-independent functional data clustering](https://mlsanity.com/paper/2602.22823/) (2026)
- [Neural Operators Can Discover Functional Clusters](https://mlsanity.com/paper/2602.23528/) (2026)
- [Geometric Feature Learning for Functional Data Valued on the Symmetric Positive Definite Manifold](https://mlsanity.com/paper/2609.30487/) (2026)
- [Functional Autoencoders for Amplitude-Phase Representation Learning](https://mlsanity.com/paper/2609.34207/) (2026)

## Abstract (arXiv)

We introduce FAEclust, a novel functional autoencoder framework for cluster analysis of multi-dimensional functional data, data that are random realizations of vector-valued random functions. Our framework features a universal-approximator encoder that captures complex nonlinear interdependencies among component functions, and a universal-approximator decoder capable of accurately reconstructing both Euclidean and manifold-valued functional data. Stability and robustness are enhanced through innovative regularization strategies applied to functional weights and biases. Additionally, we incorporate a clustering loss into the network's training objective, promoting the learning of latent representations that are conducive to effective clustering. A key innovation is our shape-informed clustering objective, ensuring that the clustering results are resistant to phase variations in the functions. We establish the universal approximation property of our non-linear decoder and validate the effectiveness of our model through extensive experiments.