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In Statistical Relational Artificial Intelligence, a branch of AI and machine learning which combines the logical and statistical schools of AI, one uses the concept {\em para\-metrized probabilistic graphical model (PPGM)} to model (conditional) dependencies between random variables and to make probabilistic inferences about events on a space of "possible worlds". The set of possible worlds with underlying domain $D$ (a set of objects) can be represented by the set $\mathbf{W}_D$ of all first-order structures (for a suitable signature) with domain $D$. Using a formal logic we can describe events on $\mathbf{W}_D$. By combining a logic and a PPGM we can also define a probability distribution $\mathbb{P}_D$ on $\mathbf{W}_D$ and use it to compute the probability of an event. We consider a logic, denoted $PLA$, with truth values in the unit interval, which uses aggregation functions, such as arithmetic mean, geometric mean, maximum and minimum instead of quantifiers. However we face the problem of computational efficiency and this problem is an obstacle to the wider use of methods from Statistical Relational AI in practical applications. We address this problem by proving that the described probability will, under certain assumptions on the PPGM and the sentence $\varphi$, converge as the size of $D$ tends to infinity. The convergence result is obtained by showing that every formula $\varphi(x_1, \ldots, x_k)$ which contains only "admissible" aggregation functions (e.g. arithmetic and geometric mean, max and min) is asymptotically equivalent to a formula $\psi(x_1, \ldots, x_k)$ without aggregation functions.

The monotone variational inequality is a central problem in mathematical programming that unifies and generalizes many important settings such as smooth convex optimization, two-player zero-sum games, convex-concave saddle point problems, etc. The extragradient method by Korpelevich [1976] is one of the most popular methods for solving monotone variational inequalities. Despite its long history and intensive attention from the optimization and machine learning community, the following major problem remains open. What is the last-iterate convergence rate of the extragradient method for monotone and Lipschitz variational inequalities with constraints? We resolve this open problem by showing a tight $O\left(\frac{1}{\sqrt{T}}\right)$ last-iterate convergence rate for arbitrary convex feasible sets, which matches the lower bound by Golowich et al. [2020]. Our rate is measured in terms of the standard gap function. The technical core of our result is the monotonicity of a new performance measure -- the tangent residual, which can be viewed as an adaptation of the norm of the operator that takes the local constraints into account. To establish the monotonicity, we develop a new approach that combines the power of the sum-of-squares programming with the low dimensionality of the update rule of the extragradient method. We believe our approach has many additional applications in the analysis of iterative methods.

Numerical solution of heterogeneous Helmholtz problems presents various computational challenges, with descriptive theory remaining out of reach for many popular approaches. Robustness and scalability are key for practical and reliable solvers in large-scale applications, especially for large wave number problems. In this work we explore the use of a GenEO-type coarse space to build a two-level additive Schwarz method applicable to highly indefinite Helmholtz problems. Through a range of numerical tests on a 2D model problem, discretised by finite elements on pollution-free meshes, we observe robust convergence, iteration counts that do not increase with the wave number, and good scalability of our approach. We further provide results showing a favourable comparison with the DtN coarse space. Our numerical study shows promise that our solver methodology can be effective for challenging heterogeneous applications.

We study an implicit finite-volume scheme for non-linear, non-local aggregation-diffusion equations which exhibit a gradient-flow structure, recently introduced by Bailo, Carrillo, and Hu (2020). Crucially, this scheme keeps the dissipation property of an associated fully discrete energy, and does so unconditionally with respect to the time step. Our main contribution in this work is to show the convergence of the method under suitable assumptions on the diffusion functions and potentials involved.

Graph Convolutional Network (GCN) has achieved extraordinary success in learning effective task-specific representations of nodes in graphs. However, regarding Heterogeneous Information Network (HIN), existing HIN-oriented GCN methods still suffer from two deficiencies: (1) they cannot flexibly explore all possible meta-paths and extract the most useful ones for a target object, which hinders both effectiveness and interpretability; (2) they often need to generate intermediate meta-path based dense graphs, which leads to high computational complexity. To address the above issues, we propose an interpretable and efficient Heterogeneous Graph Convolutional Network (ie-HGCN) to learn the representations of objects in HINs. It is designed as a hierarchical aggregation architecture, i.e., object-level aggregation first, followed by type-level aggregation. The novel architecture can automatically extract useful meta-paths for each object from all possible meta-paths (within a length limit), which brings good model interpretability. It can also reduce the computational cost by avoiding intermediate HIN transformation and neighborhood attention. We provide theoretical analysis about the proposed ie-HGCN in terms of evaluating the usefulness of all possible meta-paths, its connection to the spectral graph convolution on HINs, and its quasi-linear time complexity. Extensive experiments on three real network datasets demonstrate the superiority of ie-HGCN over the state-of-the-art methods.

Since real-world objects and their interactions are often multi-modal and multi-typed, heterogeneous networks have been widely used as a more powerful, realistic, and generic superclass of traditional homogeneous networks (graphs). Meanwhile, representation learning (\aka~embedding) has recently been intensively studied and shown effective for various network mining and analytical tasks. In this work, we aim to provide a unified framework to deeply summarize and evaluate existing research on heterogeneous network embedding (HNE), which includes but goes beyond a normal survey. Since there has already been a broad body of HNE algorithms, as the first contribution of this work, we provide a generic paradigm for the systematic categorization and analysis over the merits of various existing HNE algorithms. Moreover, existing HNE algorithms, though mostly claimed generic, are often evaluated on different datasets. Understandable due to the application favor of HNE, such indirect comparisons largely hinder the proper attribution of improved task performance towards effective data preprocessing and novel technical design, especially considering the various ways possible to construct a heterogeneous network from real-world application data. Therefore, as the second contribution, we create four benchmark datasets with various properties regarding scale, structure, attribute/label availability, and \etc.~from different sources, towards handy and fair evaluations of HNE algorithms. As the third contribution, we carefully refactor and amend the implementations and create friendly interfaces for 13 popular HNE algorithms, and provide all-around comparisons among them over multiple tasks and experimental settings.

Recent years have witnessed the emerging success of graph neural networks (GNNs) for modeling structured data. However, most GNNs are designed for homogeneous graphs, in which all nodes and edges belong to the same types, making them infeasible to represent heterogeneous structures. In this paper, we present the Heterogeneous Graph Transformer (HGT) architecture for modeling Web-scale heterogeneous graphs. To model heterogeneity, we design node- and edge-type dependent parameters to characterize the heterogeneous attention over each edge, empowering HGT to maintain dedicated representations for different types of nodes and edges. To handle dynamic heterogeneous graphs, we introduce the relative temporal encoding technique into HGT, which is able to capture the dynamic structural dependency with arbitrary durations. To handle Web-scale graph data, we design the heterogeneous mini-batch graph sampling algorithm---HGSampling---for efficient and scalable training. Extensive experiments on the Open Academic Graph of 179 million nodes and 2 billion edges show that the proposed HGT model consistently outperforms all the state-of-the-art GNN baselines by 9%--21% on various downstream tasks.

Graph representation learning is to learn universal node representations that preserve both node attributes and structural information. The derived node representations can be used to serve various downstream tasks, such as node classification and node clustering. When a graph is heterogeneous, the problem becomes more challenging than the homogeneous graph node learning problem. Inspired by the emerging information theoretic-based learning algorithm, in this paper we propose an unsupervised graph neural network Heterogeneous Deep Graph Infomax (HDGI) for heterogeneous graph representation learning. We use the meta-path structure to analyze the connections involving semantics in heterogeneous graphs and utilize graph convolution module and semantic-level attention mechanism to capture local representations. By maximizing local-global mutual information, HDGI effectively learns high-level node representations that can be utilized in downstream graph-related tasks. Experiment results show that HDGI remarkably outperforms state-of-the-art unsupervised graph representation learning methods on both classification and clustering tasks. By feeding the learned representations into a parametric model, such as logistic regression, we even achieve comparable performance in node classification tasks when comparing with state-of-the-art supervised end-to-end GNN models.

Recommender System (RS) is a hot area where artificial intelligence (AI) techniques can be effectively applied to improve performance. Since the well-known Netflix Challenge, collaborative filtering (CF) has become the most popular and effective recommendation method. Despite their success in CF, various AI techniques still have to face the data sparsity and cold start problems. Previous works tried to solve these two problems by utilizing auxiliary information, such as social connections among users and meta-data of items. However, they process different types of information separately, leading to information loss. In this work, we propose to utilize Heterogeneous Information Network (HIN), which is a natural and general representation of different types of data, to enhance CF-based recommending methods. HIN-based recommender systems face two problems: how to represent high-level semantics for recommendation and how to fuse the heterogeneous information to recommend. To address these problems, we propose to applying meta-graph to HIN-based RS and solve the information fusion problem with a "matrix factorization (MF) + factorization machine (FM)" framework. For the "MF" part, we obtain user-item similarity matrices from each meta-graph and adopt low-rank matrix approximation to get latent features for both users and items. For the "FM" part, we propose to apply FM with Group lasso (FMG) on the obtained features to simultaneously predict missing ratings and select useful meta-graphs. Experimental results on two large real-world datasets, i.e., Amazon and Yelp, show that our proposed approach is better than that of the state-of-the-art FM and other HIN-based recommending methods.