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1. GLAD: Learning sparse graph recovery

   Shrivastava, Harsh; Chen, Xinshi; Chen, Binghong; Lan, Guanghui; Aluru, Srinivas; Liu, Han; Song, Le (January 2020, International Conference on Learning Representations (ICLR))

   Recovering sparse conditional independence graphs from data is a fundamental problem in machine learning with wide applications. A popular formulation of the problem is an L1 regularized maximum likelihood estimation. Many convex optimization algorithms have been designed to solve this formulation to recover the graph structure. Recently, there is a surge of interest to learn algorithms directly based on data, and in this case, learn to map empirical covariance to the sparse precision matrix. However, it is a challenging task in this case, since the symmetric positive definiteness (SPD) and sparsity of the matrix are not easy to enforce in learned algorithms, and a direct mapping from data to precision matrix may contain many parameters. We propose a deep learning architecture, GLAD, which uses an Alternating Minimization (AM) algorithm as our model inductive bias, and learns the model parameters via supervised learning. We show that GLAD learns a very compact and effective model for recovering sparse graphs from data. 

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2. A Collective Learning Framework to Boost GNN Expressiveness for Node Classification

   Hang, Mengyue; Neville, Jennifer; Ribeiro, Bruno (July 2021, International Conference on Machine Learning)

   Graph Neural Networks (GNNs) have recently been used for node and graph classification tasks with great success, but GNNs model dependencies among the attributes of nearby neighboring nodes rather than dependencies among observed node labels. In this work, we consider the task of inductive node classification using GNNs in supervised and semi-supervised settings, with the goal of incorporating label dependencies. Because current GNNs are not universal (i.e., most-expressive) graph representations, we propose a general collective learning approach to increase the representation power of any existing GNN. Our framework combines ideas from collective classification with self-supervised learning, and uses a Monte Carlo approach to sampling embeddings for inductive learning across graphs. We evaluate performance on five real-world network datasets and demonstrate consistent, significant improvement in node classification accuracy, for a variety of state-of-the-art GNNs. 

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3. Graph Structure Learning with Interpretable Bayesian Neural Networks

   Wasserman, Max; Mateos, Gonzalo (September 2024, Transactions on machine learning research)

   Graphs serve as generic tools to encode the underlying relational structure of data. Often this graph is not given, and so the task of inferring it from nodal observations becomes important. Traditional approaches formulate a convex inverse problem with a smoothness promoting objective and rely on iterative methods to obtain a solution. In supervised settings where graph labels are available, one can unroll and truncate these iterations into a deep network that is trained end-to-end. Such a network is parameter efficient and inherits inductive bias from the optimization formulation, an appealing aspect for data constrained settings in, e.g., medicine, finance, and the natural sciences. But typically such settings care equally about \textit{uncertainty} over edge predictions, not just point estimates. Here we introduce novel iterations with independently interpretable parameters, i.e., parameters whose values - independent of other parameters' settings - proportionally influence characteristics of the estimated graph, such as edge sparsity. After unrolling these iterations, prior knowledge over such graph characteristics shape prior distributions} over these independently interpretable network parameters to yield a Bayesian neural network (BNN) capable of graph structure learning (GSL) from smooth signal observations. Fast execution and parameter efficiency allow for high-fidelity posterior approximation via Markov Chain Monte Carlo (MCMC) and thus uncertainty quantification on edge predictions. Informative priors unlock modeling tools from Bayesian statistics like prior predictive checks. Synthetic and real data experiments corroborate this model's ability to provide well-calibrated estimates of uncertainty, in test cases that include unveiling economic sector modular structure from S&P500 data and recovering pairwise digit similarities from MNIST images. Overall, this framework enables GSL in modest-scale applications where uncertainty on the data structure is paramount 

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4. Weak Supervision Network Embedding for Constrained Graph Learning

   Guo, Ting; Zhu, Xingquan; Wang, Yang; Chen, Fang (May 2021, Proc. of the Advances in Knowledge Discovery and Data Mining. PAKDD 2021. Lecture Notes in Computer Science)

   Karlapalem, Kamal; Cheng, Hong; Ramakrishnan, Naren; null; null; Reddy, P. Krishna; Srivastava, Jaideep; Chakraborty, Tanmoy (Ed.)

   Constrained learning, a weakly supervised learning task, aims to incorporate domain constraints to learn models without requiring labels for each instance. Because weak supervision knowledge is useful and easy to obtain, constrained learning outperforms unsupervised learning in performance and is preferable than supervised learning in terms of labeling costs. To date, constrained learning, especially constrained clustering, has been extensively studied, but was primarily focused on data in the Euclidean space. In this paper, we propose a weak supervision network embedding (WSNE) for constrained learning of graphs. Because no label is available for individual nodes, we propose a new loss function to quantify the constraint-based loss, and integrate this loss in a graph convolutional neural network (GCN) and variational graph auto-encoder (VGAE) combined framework to jointly model graph structures and node attributes. The joint optimization allows WSNE to learn embedding not only preserving network topology and content, but also satisfying the constraints. Experiments show that WSNE outperforms baselines for constrained graph learning tasks, including constrained graph clustering and constrained graph classification. 

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5. Hyperbolic Graph Convolutional Neural Networks

   Chami, I.; Ying, R.; Ré, C.; Leskovec, J. (December 2019, Advances in neural information processing systems)

   Graph convolutional neural networks (GCNs) embed nodes in a graph into Euclidean space, which has been shown to incur a large distortion when embedding real-world graphs with scale-free or hierarchical structure. Hyperbolic geometry offers an exciting alternative, as it enables embeddings with much smaller distortion. However, extending GCNs to hyperbolic geometry presents several unique challenges because it is not clear how to define neural network operations, such as feature transformation and aggregation, in hyperbolic space. Furthermore, since input features are often Euclidean, it is unclear how to transform the features into hyperbolic embeddings with the right amount of curvature. Here we propose Hyperbolic Graph Convolutional Neural Network (HGCN), the first inductive hyperbolic GCN that leverages both the expressiveness of GCNs and hyperbolic geometry to learn inductive node representations for hierarchical and scale-free graphs. We derive GCNs operations in the hyperboloid model of hyperbolic space and map Euclidean input features to embeddings in hyperbolic spaces with different trainable curvature at each layer. Experiments demonstrate that HGCN learns embeddings that preserve hierarchical structure, and leads to improved performance when compared to Euclidean analogs, even with very low dimensional embeddings: compared to state-of-the-art GCNs, HGCN achieves an error reduction of up to 63.1% in ROC AUC for link prediction and of up to 47.5% in F1 score for node classification, also improving state-of-the art on the PubMed dataset. 

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