Alignment Matters Inside and Out in Equivariant Graph Flow Matching
Abstract
Graphs are invariant under node permutations, motivating permutation-equivariant architectures in generative models. In flow matching, however, symmetry also affects the source-target coupling: one must choose both which graphs to pair and which node representatives to align. We study these two forms of alignment, termed outer and inner alignment, and their effect on equivariant graph flow matching. We connect inner alignment to transport on the graph quotient space, whose Euclidean quotient metric coincides with Gromov-Monge distance, and show that quotient couplings admit aligned representative lifts while symmetrization yields equivariant flow-matching minimizers, including for categorical endpoints. In practice, we compare random augmentation, permutation-blind minibatch optimal transport, approximate Gromov-Wasserstein alignment, and combinations of inner and outer alignment. Across continuous graph and molecular generation, alignment can substantially simplify trajectories and improve few-step generation, while its benefits depend on the alignment and computational budget. Our results highlight that effective graph flow matching benefits from alignment both inside and across graph pairs.
Related
- Intrinsic Geometry Matching: learning observation-equivariant process geometry
- CorrFlow: Exact and Permutation-Equivariant Flow Matching on the Correlation Elliptope
- Graph Matching Relaxations and Amortization for Supervised Graph Prediction
- Regularized Coupling Maps on Matrix Polytopes: From Assignment to Graph Matching and Optimal Transport
- Auditing Bayesian Graph Alignment: Diagnostic Comparisons and Reference Failure
BibTeX
@misc{piening2026alignment,
title = {{Alignment Matters Inside and Out in Equivariant Graph Flow Matching}},
author = {Moritz Piening and Christian Wald},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.26961},
eprint = {2608.26961},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.26961},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Alignment Matters Inside and Out in Equivariant Graph Flow Matching
AU - Moritz Piening
AU - Christian Wald
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2608.26961
UR - https://arxiv.org/abs/2608.26961
AB - Graphs are invariant under node permutations, motivating permutation-equivariant architectures in generative models. In flow matching, however, symmetry also affects the source-target coupling: one must choose both which graphs to pair and which node representatives to align. We study these two forms of alignment, termed outer and inner alignment, and their effect on equivariant graph flow matching. We connect inner alignment to transport on the graph quotient space, whose Euclidean quotient metric coincides with Gromov-Monge distance, and show that quotient couplings admit aligned representative lifts while symmetrization yields equivariant flow-matching minimizers, including for categorical endpoints. In practice, we compare random augmentation, permutation-blind minibatch optimal transport, approximate Gromov-Wasserstein alignment, and combinations of inner and outer alignment. Across continuous graph and molecular generation, alignment can substantially simplify trajectories and improve few-step generation, while its benefits depend on the alignment and computational budget. Our results highlight that effective graph flow matching benefits from alignment both inside and across graph pairs.
ER -