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NeuralFLoC: Neural Flow-Based Joint Registration and Clustering of Functional Data

Xinyang Xiong, Siyuan jiang, Pengcheng Zeng

TL;DR

A fully unsupervised, end-to-end deep learning framework for joint functional registration and clustering based on Neural ODE-driven diffeomorphic flows and spectral clustering that establishes universal approximation guarantees and asymptotic consistency for the proposed framework.

Abstract

Clustering functional data in the presence of phase variation is challenging, as temporal misalignment can obscure intrinsic shape differences and degrade clustering performance. Most existing approaches treat registration and clustering as separate tasks or rely on restrictive parametric assumptions. We present \textbf{NeuralFLoC}, a fully unsupervised, end-to-end deep learning framework for joint functional registration and clustering based on Neural ODE-driven diffeomorphic flows and spectral clustering. The proposed model learns smooth, invertible warping functions and cluster-specific templates simultaneously, effectively disentangling phase and amplitude variation. We establish universal approximation guarantees and asymptotic consistency for the proposed framework. Experiments on functional benchmarks show state-of-the-art performance in both registration and clustering, with robustness to missing data, irregular sampling, and noise, while maintaining scalability. Code is available at https://anonymous.4open.science/r/NeuralFLoC-FEC8.

NeuralFLoC: Neural Flow-Based Joint Registration and Clustering of Functional Data

TL;DR

A fully unsupervised, end-to-end deep learning framework for joint functional registration and clustering based on Neural ODE-driven diffeomorphic flows and spectral clustering that establishes universal approximation guarantees and asymptotic consistency for the proposed framework.

Abstract

Clustering functional data in the presence of phase variation is challenging, as temporal misalignment can obscure intrinsic shape differences and degrade clustering performance. Most existing approaches treat registration and clustering as separate tasks or rely on restrictive parametric assumptions. We present \textbf{NeuralFLoC}, a fully unsupervised, end-to-end deep learning framework for joint functional registration and clustering based on Neural ODE-driven diffeomorphic flows and spectral clustering. The proposed model learns smooth, invertible warping functions and cluster-specific templates simultaneously, effectively disentangling phase and amplitude variation. We establish universal approximation guarantees and asymptotic consistency for the proposed framework. Experiments on functional benchmarks show state-of-the-art performance in both registration and clustering, with robustness to missing data, irregular sampling, and noise, while maintaining scalability. Code is available at https://anonymous.4open.science/r/NeuralFLoC-FEC8.
Paper Structure (53 sections, 2 theorems, 31 equations, 8 figures, 3 tables, 2 algorithms)

This paper contains 53 sections, 2 theorems, 31 equations, 8 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Related Work
  4. Curve Registration and the SRVF Framework.
  5. Neural Ordinary Differential Equations.
  6. Functional Clustering.
  7. Contributions
  8. Preliminaries
  9. Warping Space and Diffeomorphisms.
  10. Neural ODEs.
  11. SRVF.
  12. Methodology
  13. Latent Initialization via 1D-CNN
  14. Continuous Time-Warping via Neural ODEs
  15. Clustering Assignment in Spectral Domain
  16. ...and 38 more sections

Key Result

Theorem 4.1

Let $\Gamma$ denote the space of boundary-preserving diffeomorphisms on $[0,1]$. Assume that: Then, for any $\varepsilon>0$, there exist parameters $(\Theta_{\mathrm{enc}},\Theta_{\mathrm{ode}})$ such that the induced warping functions $\hat{\gamma}_i$ satisfy where $R(\gamma)$ denotes the total SRVF-based registration loss.

Figures (8)

  • Figure 1: Overview of NeuralFLoC. A 1D-CNN encodes raw functional data. A Neural ODE generates smooth, invertible warping functions $\gamma_i(t)$ to align curves, and the aligned features are clustered via a Fourier-based soft-assignment module. The model is trained end-to-end for joint registration and clustering.
  • Figure 2: (a) Raw data in Shapes dataset with two ground-truth groups indicated by black and grey. (b) Learned warping functions $\gamma(t)$'s by Ours. (c) Alignment by Ours. (d) Alignment by Ours (N-ODE $\rightarrow$ SrvfRegNet). Colored curves in (b)-(d) indicate the clustering results.
  • Figure 3: Left: Sensitivity to loss weight $\alpha$. Right: Sensitivity to number of basis functions $K$
  • Figure 4: Robustness of NeuralFLoC to missing data, irregular sampling, and additive noise (left to right).
  • Figure A.5: (a) Raw data in Wave ($d=1$) dataset with two ground-truth groups indicated by black and grey. (b) Learned warping functions $\gamma(t)$'s by Ours. (c) Alignment by Ours. (d) Alignment by Ours (N-ODE $\rightarrow$ SrvfRegNet). Colored curves in (b)-(d) indicate the clustering results.
  • ...and 3 more figures

Theorems & Definitions (4)

  • Theorem 4.1: Approximation Consistency of Neural Registration
  • Theorem 4.2: Consistency of Joint Registration and Clustering
  • proof : Proof of Theorem Theorem \ref{['thm:approx']}
  • proof : Proof of Theorem \ref{['thm:ConsJRC']}