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

2025-09-26

AI Summary

This paper addresses shape-aware clustering of multivariate functional data—particularly vector-valued random functions exhibiting complex nonlinear dependencies and phase variation. We propose FAEclust, a deep functional autoencoder-based framework that jointly optimizes clustering and reconstruction losses to learn discriminative latent representations, explicitly modeling nonlinear inter-component dependencies. To enhance robustness against phase variation, we introduce a shape-aware clustering objective grounded in functional alignment principles. Additionally, we design a functional-parameter regularization scheme to improve model stability. Theoretically, we prove that the decoder possesses universal approximation capability for continuous functional mappings. Empirical evaluations on both Euclidean- and manifold-valued functional datasets demonstrate that FAEclust significantly outperforms state-of-the-art methods in clustering accuracy and generalization performance, achieving superior robustness to amplitude-phase separation challenges.

Technology Category

• Machine Learning: Clustering
• Computer Vision: Representation Learning for Vision
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

Abstract

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.

Problem

• Clustering multi-dimensional functional data with nonlinear interdependencies
• Enhancing clustering robustness against functional phase variations
• Developing deep autoencoder framework for shape-informed cluster analysis

Innovation

• Deep functional autoencoders cluster multi-dimensional data
• Regularization enhances stability of functional weights
• Shape-informed clustering resists phase variations