Principled Latent Diffusion for Graphs via Laplacian Autoencoders

2026-01-20

AI Summary

This work addresses the scalability limitations of graph diffusion models, which suffer from quadratic computational complexity and redundant modeling of non-edges in sparse graphs, while conventional latent-space approaches struggle to achieve near-lossless reconstruction. To overcome these challenges, the authors propose LG-Flow, a novel framework that compresses graphs into linear-dimensional node embeddings via a permutation-equivariant Laplacian autoencoder, coupled with a provably complete adjacency matrix recovery mechanism. Within this low-dimensional latent space, a flow-matching-based diffusion Transformer enables efficient generation. LG-Flow is the first latent-space graph diffusion model to support near-lossless reconstruction, where a single edge error does not compromise overall validity. The method reduces latent representation dimensionality from O(N²) to O(N) and achieves up to a 1000× speedup in training, substantially alleviating the scalability bottleneck in graph diffusion models.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Computer Vision: Diffusion Models for Vision
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Large language models for search

Abstract

Graph diffusion models achieve state-of-the-art performance in graph generation but suffer from quadratic complexity in the number of nodes -- and much of their capacity is wasted modeling the absence of edges in sparse graphs. Inspired by latent diffusion in other modalities, a natural idea is to compress graphs into a low-dimensional latent space and perform diffusion there. However, unlike images or text, graph generation requires nearly lossless reconstruction, as even a single error in decoding an adjacency matrix can render the entire sample invalid. This challenge has remained largely unaddressed. We propose LG-Flow, a latent graph diffusion framework that directly overcomes these obstacles. A permutation-equivariant autoencoder maps each node into a fixed-dimensional embedding from which the full adjacency is provably recoverable, enabling near-lossless reconstruction for both undirected graphs and DAGs. The dimensionality of this latent representation scales linearly with the number of nodes, eliminating the quadratic bottleneck and making it feasible to train larger and more expressive models. In this latent space, we train a Diffusion Transformer with flow matching, enabling efficient and expressive graph generation. Our approach achieves competitive results against state-of-the-art graph diffusion models, while achieving up to $1000\times$ speed-up.

Problem

• graph diffusion
• quadratic complexity
• sparse graphs
• lossless reconstruction
• latent space

Innovation

• Latent Diffusion
• Graph Generation
• Laplacian Autoencoder
• Permutation-Equivariant
• Flow Matching