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We present a novel spectral embedding of graphs that incorporates weights assigned to the nodes, quantifying their relative importance. This spectral embedding is based on the first eigenvectors of some properly normalized version of the Laplacian. We prove that these eigenvectors correspond to the configurations of lowest energy of an equivalent physical system, either mechanical or electrical, in which the weight of each node can be interpreted as its mass or its capacitance, respectively. Experiments on a real dataset illustrate the impact of weighting on the embedding.
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Learning useful representations is a key ingredient to the success of modern machine learning. Currently, representation learning mostly relies on embedding data into Euclidean space. However, recent work has shown that data in some domains is better modeled by non-euclidean metric spaces, and inappropriate geometry can result in inferior performance. In this paper, we aim to eliminate the inductive bias imposed by the embedding space geometry. Namely, we propose to map data into more general non-vector metric spaces: a weighted graph with a shortest path distance. By design, such graphs can model arbitrary geometry with a proper configuration of edges and weights. Our main contribution is PRODIGE: a method that learns a weighted graph representation of data end-to-end by gradient descent. Greater generality and fewer model assumptions make PRODIGE more powerful than existing embedding-based approaches. We confirm the superiority of our method via extensive experiments on a wide range of tasks, including classification, compression, and collaborative filtering.
This paper revisits spectral graph convolutional neural networks (graph-CNNs) given in Defferrard (2016) and develops the Laplace-Beltrami CNN (LB-CNN) by replacing the graph Laplacian with the LB operator. We then define spectral filters via the LB operator on a graph. We explore the feasibility of Chebyshev, Laguerre, and Hermite polynomials to approximate LB-based spectral filters and define an update of the LB operator for pooling in the LBCNN. We employ the brain image data from Alzheimers Disease Neuroimaging Initiative (ADNI) and demonstrate the use of the proposed LB-CNN. Based on the cortical thickness of the ADNI dataset, we showed that the LB-CNN didnt improve classification accuracy compared to the spectral graph-CNN. The three polynomials had a similar computational cost and showed comparable classification accuracy in the LB-CNN or spectral graph-CNN. Our findings suggest that even though the shapes of the three polynomials are different, deep learning architecture allows us to learn spectral filters such that the classification performance is not dependent on the type of the polynomials or the operators (graph Laplacian and LB operator).
Knowledge graph embedding plays an important role in knowledge representation, reasoning, and data mining applications. However, for multiple cross-domain knowledge graphs, state-of-the-art embedding models cannot make full use of the data from different knowledge domains while preserving the privacy of exchanged data. In addition, the centralized embedding model may not scale to the extensive real-world knowledge graphs. Therefore, we propose a novel decentralized scalable learning framework, emph{Federated Knowledge Graphs Embedding} (FKGE), where embeddings from different knowledge graphs can be learnt in an asynchronous and peer-to-peer manner while being privacy-preserving. FKGE exploits adversarial generation between pairs of knowledge graphs to translate identical entities and relations of different domains into near embedding spaces. In order to protect the privacy of the training data, FKGE further implements a privacy-preserving neural network structure to guarantee no raw data leakage. We conduct extensive experiments to evaluate FKGE on 11 knowledge graphs, demonstrating a significant and consistent improvement in model quality with at most 17.85% and 7.90% increases in performance on triple classification and link prediction tasks.
Detecting communities on graphs has received significant interest in recent literature. Current state-of-the-art community embedding approach called textit{ComE} tackles this problem by coupling graph embedding with community detection. Considering the success of hyperbolic representations of graph-structured data in last years, an ongoing challenge is to set up a hyperbolic approach for the community detection problem. The present paper meets this challenge by introducing a Riemannian equivalent of textit{ComE}. Our proposed approach combines hyperbolic embeddings with Riemannian K-means or Riemannian mixture models to perform community detection. We illustrate the usefulness of this framework through several experiments on real-world social networks and comparisons with textit{ComE} and recent hyperbolic-based classification approaches.
Many real-world graphs involve different types of nodes and relations between nodes, being heterogeneous by nature. The representation learning of heterogeneous graphs (HGs) embeds the rich structure and semantics of such graphs into a low-dimensional space and facilitates various data mining tasks, such as node classification, node clustering, and link prediction. In this paper, we propose a self-supervised method that learns HG representations by relying on knowledge exchange and discovery among different HG structural semantics (meta-paths). Specifically, by maximizing the mutual information of meta-path representations, we promote meta-path information fusion and consensus, and ensure that globally shared semantics are encoded. By extensive experiments on node classification, node clustering, and link prediction tasks, we show that the proposed self-supervision both outperforms and improves competing methods by 1% and up to 10% for all tasks.
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