Graph matching has important applications in pattern recognition and beyond. Current approaches predominantly adopt supervised learning, demanding extensive labeled data which can be limited or costly. Meanwhile, self-supervised learning methods for graph matching often require additional side information such as extra categorical information and input features, limiting their application to the general case. Moreover, designing the optimal graph augmentations for self-supervised graph matching presents another challenge to ensure robustness and efficacy. To address these issues, we introduce a novel Graph-centric Contrastive framework for Graph Matching (GCGM), capitalizing on a vast pool of graph augmentations for contrastive learning, yet without needing any side information. Given the variety of augmentation choices, we further introduce a Boosting-inspired Adaptive Augmentation Sampler (BiAS), which adaptively selects more challenging augmentations tailored for graph matching. Through various experiments, our GCGM surpasses state-of-the-art self-supervised methods across various datasets, marking a significant step toward more effective, efficient and general graph matching.

The conditional gradient method (CGM) has been widely used for fast sparse approximation, having a low per iteration computational cost for structured sparse regularizers. We explore the sparsity acquiring properties of a generalized CGM (gCGM), where the constraint is replaced by a penalty function based on a gauge penalty; this can be done without significantly increasing the per-iteration computation, and applies to general notions of sparsity. Without assuming bounded iterates, we show $O(1/t)$ convergence of the function values and gap of gCGM. We couple this with a safe screening rule, and show that at a rate $O(1/(t\delta^2))$, the screened support matches the support at the solution, where $\delta \geq 0$ measures how close the problem is to being degenerate. In our experiments, we show that the gCGM for these modified penalties have similar feature selection properties as common penalties, but with potentially more stability over the choice of hyperparameter.

The Collective Graphical Model (CGM) models a population of independent and identically distributed individuals when only collective statistics (i.e., counts of individuals) are observed. Exact inference in CGMs is intractable, and previous work has explored Markov Chain Monte Carlo (MCMC) and MAP approximations for learning and inference. This paper studies Gaussian approximations to the CGM. As the population grows large, we show that the CGM distribution converges to a multivariate Gaussian distribution (GCGM) that maintains the conditional independence properties of the original CGM. If the observations are exact marginals of the CGM or marginals that are corrupted by Gaussian noise, inference in the GCGM approximation can be computed efficiently in closed form. If the observations follow a different noise model (e.g., Poisson), then expectation propagation provides efficient and accurate approximate inference. The accuracy and speed of GCGM inference is compared to the MCMC and MAP methods on a simulated bird migration problem. The GCGM matches or exceeds the accuracy of the MAP method while being significantly faster.