PolarFlow: Spectral Geometric Flow Matching for Directed Graph Generation via Polar Decomposition

Abstract

Directed graphs model asymmetric relationships in biology, transportation, social networks, and electronic design automation. Recent discrete diffusion and flow-matching methods generate directed graphs effectively, but their edge-wise categorical formulations require dense predictions over all ordered node pairs and repeated categorical updates during generation. This motivates a fundamental question: can comparable generative quality be achieved through a more computationally efficient formulation that better exploits the structure of directed adjacency matrices? Existing benchmarks also cover limited network structures, restricting comprehensive evaluation. We introduce PolarFlow, a continuous geometric flow-matching framework based on polar decomposition and truncated spectral coordinates. Its representation comprises four coupled components: a Stiefel frame and rotation angles encoding the orthogonal factor, and a second Stiefel frame and nonnegative eigenvalues characterizing the positive-semidefinite factor. PolarFlow learns a joint velocity field along geometry-aware conditional paths on this product space, reconstructs a continuous adjacency-score matrix, and discretizes it into a binary directed graph. By replacing discrete edge-wise generation with joint continuous transport, PolarFlow improves computational efficiency while retaining global adjacency structure. To broaden evaluation, we introduce complementary Circuit and Twitter benchmarks spanning engineered functional and naturally evolved social networks. Across synthetic and real-world datasets, PolarFlow achieves generative quality comparable to competitive discrete methods while substantially reducing training and sampling time.