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Riemannian Flow Matching for Disentangled Graph Domain Adaptation

Yingxu Wang, Xinwang Liu, Mengzhu Wang, Siyang Gao, Nan Yin

TL;DR

Riemannian Flow Matching for Disentangled Graph Domain Adaptation (DisRFM) tackles cross-domain graph classification by marrying geometry-aware representations with stable transport. By embedding graphs on a constant-curvature Riemannian manifold in polar coordinates, it disentangles structure (radius $r$) from semantics (angle $\theta$), preserving topology via radial Wasserstein alignment and enforcing semantic discriminability via angular clustering. It replaces unstable adversarial alignment with a stable, geodesic-flow transport on the manifold, supported by Lyapunov stability and a tighter target-risk bound. Empirically, DisRFM achieves state-of-the-art results across diverse domain shifts while exhibiting superior training stability and interpretable manifold geometry. The framework opens avenues for adaptive curvature learning and extensions to heterogeneous graphs and generative tasks in science.

Abstract

Graph Domain Adaptation (GDA) typically uses adversarial learning to align graph embeddings in Euclidean space. However, this paradigm suffers from two critical challenges: Structural Degeneration, where hierarchical and semantic representations are entangled, and Optimization Instability, which arises from oscillatory dynamics of minimax adversarial training. To tackle these issues, we propose DisRFM, a geometry-aware GDA framework that unifies Riemannian embedding and flow-based transport. First, to overcome structural degeneration, we embed graphs into a Riemannian manifold. By adopting polar coordinates, we explicitly disentangle structure (radius) from semantics (angle). Then, we enforce topology preservation through radial Wasserstein alignment and semantic discrimination via angular clustering, thereby preventing feature entanglement and collapse. Second, we address the instability of adversarial alignment by using Riemannian flow matching. This method learns a smooth vector field to guide source features toward the target along geodesic paths, guaranteeing stable convergence. The geometric constraints further guide the flow to maintain the disentangled structure during transport. Theoretically, we prove the asymptotic stability of the flow matching and derive a tighter bound for the target risk. Extensive experiments demonstrate that DisRFM consistently outperforms state-of-the-art methods.

Riemannian Flow Matching for Disentangled Graph Domain Adaptation

TL;DR

Riemannian Flow Matching for Disentangled Graph Domain Adaptation (DisRFM) tackles cross-domain graph classification by marrying geometry-aware representations with stable transport. By embedding graphs on a constant-curvature Riemannian manifold in polar coordinates, it disentangles structure (radius ) from semantics (angle ), preserving topology via radial Wasserstein alignment and enforcing semantic discriminability via angular clustering. It replaces unstable adversarial alignment with a stable, geodesic-flow transport on the manifold, supported by Lyapunov stability and a tighter target-risk bound. Empirically, DisRFM achieves state-of-the-art results across diverse domain shifts while exhibiting superior training stability and interpretable manifold geometry. The framework opens avenues for adaptive curvature learning and extensions to heterogeneous graphs and generative tasks in science.

Abstract

Graph Domain Adaptation (GDA) typically uses adversarial learning to align graph embeddings in Euclidean space. However, this paradigm suffers from two critical challenges: Structural Degeneration, where hierarchical and semantic representations are entangled, and Optimization Instability, which arises from oscillatory dynamics of minimax adversarial training. To tackle these issues, we propose DisRFM, a geometry-aware GDA framework that unifies Riemannian embedding and flow-based transport. First, to overcome structural degeneration, we embed graphs into a Riemannian manifold. By adopting polar coordinates, we explicitly disentangle structure (radius) from semantics (angle). Then, we enforce topology preservation through radial Wasserstein alignment and semantic discrimination via angular clustering, thereby preventing feature entanglement and collapse. Second, we address the instability of adversarial alignment by using Riemannian flow matching. This method learns a smooth vector field to guide source features toward the target along geodesic paths, guaranteeing stable convergence. The geometric constraints further guide the flow to maintain the disentangled structure during transport. Theoretically, we prove the asymptotic stability of the flow matching and derive a tighter bound for the target risk. Extensive experiments demonstrate that DisRFM consistently outperforms state-of-the-art methods.
Paper Structure (40 sections, 5 theorems, 60 equations, 9 figures, 15 tables)

This paper contains 40 sections, 5 theorems, 60 equations, 9 figures, 15 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Problem Definition
  5. Riemannian Graph Convolution.
  6. Methodology
  7. Overview of Framework
  8. Disentangled Geometric Constraints
  9. Geometry-Aware Manifold Transport
  10. Learning Framework
  11. Theoretical Analysis
  12. Experiments
  13. Experimental Settings
  14. Performance Comparison
  15. Training Stability Analysis
  16. ...and 25 more sections

Key Result

Proposition 4.1

Let $\text{Vol}(\mathcal{B}_{\mathcal{X}}(R))$ denote the volume of a ball with radius $R$ in a geometric space $\mathcal{X} \in \{\mathbb{E}, \mathbb{S}, \mathbb{H}\}$. As $R \to \infty$, the volume growth behaviors diverge significantly:

Figures (9)

  • Figure 1: (a) (c) show Structural Degeneration and Optimization Instability in Euclidean adversarial alignment. (b) (d) describe the Riemannian polar coordinates that enable disentangled structure-semantics encoding, while flow matching ensures stable convergence.
  • Figure 2: Training stability comparisons.
  • Figure 3: Visualization of TSNE and Manifold Geometry.
  • Figure 4: Sensitivity analysis of manifold curvature $c$ and balance coefficient ($\lambda_1$, $\lambda_2$) on the Mutagenicity dataset.
  • Figure 5: Visualization of domain shifts across different types. (a) Node distribution shift between sub-datasets of PROTEINS. (b) Edge distribution shift between sub-datasets of PROTEINS. (c) Feature distribution shift between BZR and BZR_MD datasets.
  • ...and 4 more figures

Theorems & Definitions (5)

  • Proposition 4.1: Exponential Volume Expansion
  • Theorem 4.2: Disentanglement Efficiency via Metric Orthogonality
  • Theorem 4.3: Spectral Instability of Adversarial Training
  • Theorem 4.4: Asymptotic Stability of DisRFM
  • Theorem 4.5: Tighter Generalization Bound