Network Learning with Semi-relaxed Gromov-Wasserstein

2026-06-01

AI Summary

This work addresses the NP-hard combinatorial optimization problem arising in estimating generative mechanisms of large-scale networks when node labels are missing. To tackle this challenge, the authors propose a semi-relaxed Gromov–Wasserstein (srGW) framework that transforms the problem into a tractable low-dimensional representation via probabilistic coupling relaxation. The resulting objective function, optimized using a block-coordinate conditional gradient algorithm, achieves an optimality gap convergence rate of O(1/n) while preserving near-deterministic solution quality. Theoretical analysis establishes consistency and minimax-optimal convergence rates for both stochastic block models and Hölder-smooth graphons. Empirical evaluations on synthetic and real-world datasets demonstrate the method’s computational scalability and statistical optimality, effectively recovering the underlying network structure with high efficiency.

Abstract

Estimating the generative mechanism of large-scale networks is a fundamental challenge in statistical machine learning. It requires the identification of the latent connectivity structure, which is in general an NP-hard combinatorial problem due to the absence of canonical node labels. We address this challenge by allowing for probabilistic couplings, thereby relaxing the assignment problem. Our estimation framework can be formulated as a semi-relaxed Gromov-Wasserstein objective and provides a low-dimensional representation of the generative structure. We solve this via a block-coordinate conditional gradient algorithm. Despite the relaxation, the resulting solution is typically deterministic: in fact, we show that the optimality gap between the relaxed solution and the deterministic assignment vanishes at rate $O(1/n)$, where $n$ is the number of nodes. This allows for tractable recovery of the underlying model and enables rigorous statistical analysis: we establish consistency and minimax-optimal convergence rates for both stochastic block models and Holder-smooth graphons. Our implementation scales efficiently with $n$, as demonstrated on both synthetic and real-world datasets.

Problem

• network generative mechanism
• latent connectivity structure
• node correspondence
• graphon estimation
• stochastic block model

Innovation

• semi-relaxed Gromov-Wasserstein
• probabilistic coupling
• graphon estimation
• stochastic block model
• block-coordinate conditional gradient