metric-graph generative modeling

Existing deep generative models struggle to handle continuous probability distributions supported on compact metric graphs. This work proposes a novel approach that embeds the metric graph into a smooth ambient space—accommodating both extrinsic Euclidean and intrinsic tropical Abel–Jacobi embeddings—and solves an entropy-regularized Kantorovich problem via neural semi-dual parameterization, subsequently projecting generated samples back onto the original graph. To the best of our knowledge, this is the first method enabling deep generative modeling of probability distributions with continuous support on metric graphs, with theoretical guarantees that the generator converges weakly to the true optimal transport coupling. Experiments demonstrate that the method matches or outperforms discrete graph-based optimal transport baselines across diverse geometric graph structures and successfully scales to million-scale Manhattan Uber pickup location data.

Modeling the joint distribution of high-dimensional discrete variables (e.g., binary or categorical data) suffers from exponential computational complexity as the number of categories grows, severely limiting scalability. Method: This paper proposes a generative model based on the Random Assignment Flow (RAF), which models discrete distributions via measure transport on a statistical submanifold. It uniquely integrates *e*-connection geodesics from information geometry with conditional Riemannian flow matching, enabling simulation-free, end-to-end training. Contribution/Results: The model achieves linear time complexity in the number of affinity function parameters—bypassing the exponential barrier of conventional methods—while supporting efficient sampling and exact log-likelihood evaluation. Empirical validation on structured image annotation demonstrates superior scalability: performance degradation under increasing category count is markedly slower than state-of-the-art baselines, confirming its exceptional scalability and practical utility for large-scale discrete modeling.

Existing evaluation metrics for deep generative models (e.g., VAEs, GANs, diffusion models, Transformers) in engineering design—largely borrowed from statistical likelihood-based measures—fail to capture design-critical properties such as constraint satisfaction, functional performance, and design value. Method: We propose the first multidimensional evaluation framework tailored to engineering design, comprising four orthogonal dimensions: constraint compliance, functional effectiveness, novelty, and conditional controllability. We further develop an open-source, reproducible benchmark suite and software toolkit to bridge machine learning theory and design practice. Contribution/Results: The framework is rigorously validated on 2D visualization case studies and real-world engineering tasks—including bicycle frame and structural topology generation. Experiments demonstrate substantial improvements in alignment between automated evaluation and human-assessed design value: target achievement rate (+23.6%), geometric constraint compliance (+31.4%), and design novelty (+18.9%).

Network data often violate metric space axioms—due to incomplete and asymmetric relationships—making conventional metric embeddings inadequate. Method: We propose the first axiomatized framework for projecting networks into *q*-metric spaces, introducing two interpretable axioms that formally characterize valid projections; we rigorously prove existence and uniqueness of such mappings. Furthermore, we design a metric-tree-based approximate nearest-neighbor search paradigm that preserves theoretical uniqueness while substantially accelerating similarity retrieval over network structures. Contribution/Results: Experiments demonstrate that our method delivers high-quality approximate solutions to combinatorial optimization tasks. It establishes a novel geometric modeling paradigm for non-metric network data, enabling both theoretically grounded representation and efficient large-scale computation.

Existing graph generation models suffer from inefficiency in modeling directly in the raw space and difficulty preserving discrete graph structure when operating in continuous latent spaces. To address these challenges, we propose GLAD—the first equivariant discrete latent-space graph generative model. Our approach fundamentally departs from continuous latent-variable assumptions by constructing a strictly discrete, group-equivariant latent space via vector quantization. Second, we introduce the first adaptation of diffusion bridges as a learnable, structure-aware prior tailored to discrete graph latent spaces. Third, we employ group-equivariant neural networks within an end-to-end architecture, enabling global symmetry modeling without requiring graph decomposition. Evaluated on multiple standard graph generation benchmarks, GLAD achieves state-of-the-art performance while simultaneously guaranteeing strict equivariance, exact discreteness, and superior generative fidelity—establishing the first latent-space framework that unifies all three properties.

This work addresses the challenges of modeling complex dependencies and mitigating redundancy in high-dimensional parameter spaces for discrete data generation. It introduces, for the first time, a Riemannian geometric structure with isometric properties into the exponential parameter space of product manifolds over categorical distributions, thereby constructing a low-dimensional latent subspace. By leveraging the Riemannian metric, geodesics within this subspace become straight lines, enabling consistent and efficient flow-matching training. The proposed approach substantially reduces the dimensionality of latent variables while preserving strong representational capacity for discrete data distributions. Experimental results demonstrate that the model achieves accurate and efficient discrete data generation using a significantly lower-dimensional latent space, effectively balancing computational efficiency with modeling performance.

Existing graph generation models (GGMs) are commonly evaluated using Maximum Mean Discrepancy (MMD), which lacks sensitivity to domain-specific structural fidelity—particularly the preservation of semantically meaningful topological patterns across diverse graph domains. Method: We propose Representation-aware Graph Generation Model evaluation (RGM), the first GGM evaluation framework incorporating geometric deep learning. RGM employs trained graph classifiers to assess intra-domain structural consistency between generated and real graphs, enabling fine-grained, semantically interpretable structural fidelity assessment. Contribution/Results: Evaluated on synthetic–real hybrid benchmarks, RGM uncovers systematic structural preservation failures in state-of-the-art models (e.g., GRAN, EDGE) under cross-domain generalization. Empirical results demonstrate that RGM significantly outperforms MMD in sensitivity, robustness, and discriminative power, establishing a new, principled benchmark for rigorous GGM evaluation.

This work reformulates generative modeling from an observer’s perspective, recasting manifold learning as a decoupling problem between geometric support and probability distribution. Methodologically, it introduces a novel framework grounded in continuous percolation theory, establishing a rigorous topological isomorphism between the percolation phase transition in random geometric graphs and the underlying data manifold. A differentiable “percolation shift” metric is designed to detect structural deficiencies—such as disconnected components or topological voids—that evade detection by conventional statistical metrics (e.g., FID); this metric is jointly optimized with FID in a dual-objective loss. The approach effectively mitigates manifold collapse while substantially expanding topological diversity without sacrificing fidelity, achieving, for the first time in generative modeling, a theoretically guaranteed “super-generalization” regime. The core contribution lies in integrating percolation phase transition theory into generative modeling, thereby establishing a new geometric-topological paradigm for manifold structure assessment and optimization.