Riemannian Flow Matching for Disentangled Graph Domain Adaptation

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.

💡 Research Summary

Graph Domain Adaptation (GDA) traditionally relies on adversarial learning to align graph embeddings in Euclidean space. This paradigm suffers from two fundamental problems: (1) structural degeneration, where hierarchical graph information and semantic identity become entangled, and (2) optimization instability, caused by the oscillatory dynamics of the minimax game. In “Riemannian Flow Matching for Disentangled Graph Domain Adaptation” the authors introduce DisRFM, a novel framework that simultaneously addresses both issues by moving the entire adaptation process onto a constant‑curvature Riemannian manifold and by replacing adversarial alignment with a continuous flow‑matching formulation.

Manifold embedding and polar disentanglement
Graphs are encoded by a Riemannian Graph Convolutional Network that operates on a manifold (\mathcal{M}_c) with curvature (c) (hyperbolic, Euclidean or spherical). After mapping each graph representation to the tangent space at the origin, the vector is decomposed into a radial scalar (r = |v|_2) and a unit direction (u = v/|v|_2). The radius encodes structural hierarchy (e.g., depth, centrality) while the direction (angular component) captures semantic identity. This polar representation explicitly separates the two types of information, which is impossible in a flat Euclidean space where they are forced to share the same dimensions.

Disentangled geometric constraints
Radial alignment: The distributions of radii in source and target domains are matched using a univariate Wasserstein distance. By sorting the radii in each mini‑batch, the loss reduces to an L1 distance between order statistics, providing an efficient and exact computation. This forces the target domain to adopt the same global structural profile as the source, preventing collapse of hierarchical patterns.

Angular alignment: Class prototypes are taken from the source classifier weights. For each target sample, the angular similarity to each prototype is computed via cosine similarity of unit vectors. Because true target labels are unavailable, pseudo‑labels are generated, but only high‑confidence predictions (confidence > (\zeta)) are kept (confidence gating). Moreover, a radial reliability weight (\alpha_i = \exp(-|v_i|^2)) emphasizes samples with small radii, which are presumed more domain‑invariant. The final angular loss is a confidence‑weighted cross‑entropy with temperature scaling, encouraging tight angular clusters around class prototypes while respecting the reliability weighting.

Riemannian flow matching
Instead of a discriminator‑generator adversarial game, the authors model the domain shift as a continuous transport map on (\mathcal{M}_c). Given source point (z_S) and target point (z_T), the intermediate state at time (t\in