Event prediction tasks often handle spatio-temporal data distributed in a large spatial area. Different regions in the area exhibit different characteristics while having latent correlations. This spatial heterogeneity and correlations greatly affect the spatio-temporal distributions of event occurrences, which has not been addressed by state-of-the-art models. Learning spatial dependencies of events in a continuous space is challenging due to its fine granularity and a lack of prior knowledge. In this work, we propose a novel Graph Spatio-Temporal Point Process (GSTPP) model for fine-grained event prediction. It adopts an encoder-decoder architecture that jointly models the state dynamics of spatially localized regions using neural Ordinary Differential Equations (ODEs). The state evolution is built on the foundation of a novel Self-Adaptive Anchor Graph (SAAG) that captures spatial dependencies. By adaptively localizing the anchor nodes in the space and jointly constructing the correlation edges between them, the SAAG enhances the model’s ability of learning complex spatial event patterns. The proposed GSTPP model greatly improves the accuracy of fine-grained event prediction. Extensive experimental results show that our method greatly improves the prediction accuracy over existing spatio-temporal event prediction approaches.

Multi-view graph clustering has been intensively investigated during the past years. However, existing methods are still limited in two main aspects. On the one hand, most of them can not deal with data that have both attributes and graphs. Nowadays, multi-view attributed graph data are ubiquitous and the need for effective clustering methods is growing. On the other hand, many state-of-the-art algorithms are either shallow or deep models. Shallow methods may seriously restrict their capacity for modeling complex data, while deep approaches often involve large number of parameters and are expensive to train in terms of running time and space needed. In this paper, we propose a novel multi-view attributed graph clustering (MAGC) framework, which exploits both node attributes and graphs. Our novelty lies in three aspects. First, instead of deep neural networks, we apply a graph filtering technique to achieve a smooth node representation. Second, the original graph could be noisy or incomplete and is not directly applicable, thus we learn a consensus graph from data by considering the heterogeneous views. Third, high-order relations are explored in a flexible way by designing a new regularizer. Extensive experiments demonstrate the superiority of our method in terms of effectiveness and efficiency.

Hypergraph streams provide an efficient model to express and preserve complex connections in various applications. Triangles in a hypergraph can be formed by vertices and hyperedges. The specific counts of triangles are important to analyze various applications. Due to the huge costs of counting triangles based on the whole datasets, a sampling-and-estimating framework has low overhead while obtaining a relatively accurate result. However, existing sampling algorithms focus on pairwise graph streams, they estimate the counts of triangles formed by vertices with large estimation errors and can not be applied to count triangles formed by hyperedges. Therefore, this paper first proposes a sampling-and-estimating framework that produces hyperedge samples using a reservoir with static capacity to estimate the total counts of triangles by inferring the probabilities of forming the triangles respectively. Furthermore, to improve the estimation accuracy, this paper proposes another sampling-and-estimating framework to produce samples in the form of hyperedge pairs which can be used to compute the probabilities of the formations of triangles more accurately and then estimate the total triangle counts with smaller estimation variances. The extensive experiments based on real-world datasets confirm the efficiency and accuracy of our proposed frameworks for counting triangles in different types of hypergraphs at a small sampling ratio.

Though graph representation learning (GRL) has made significant progress, it is still a challenge to extract and embed the rich topological structure and feature information in an adequate way. Most existing methods focus on local structure and fail to fully incorporate the global topological structure. To this end, we propose a novel Structure-Preserving Graph Representation Learning (SPGRL) method, to fully capture the structure information of graphs. Specifically, to reduce the uncertainty and misinformation of the original graph, we construct a feature graph as a complementary view via k-Nearest Neighbor method. The feature graph can be used to contrast at node-level to capture the local relation. Besides, we retain the global topological structure information by maximizing the mutual information (MI) of the whole graph and feature embeddings, which is theoretically reduced to exchanging the feature embeddings of the feature and the original graphs to reconstruct themselves. Extensive experiments show that our method has quite superior performance on semi-supervised node classification task and excellent robustness under noise perturbation on graph structure or node features. The source code is available at https://github.com/uestc-lese/SPGRL.

Graph clustering is a prevalent issue associated with social networks, data mining, and machine learning; its objective is to detect communities or groups in networks. Inspired by the recent success of deep learning (DL), new DL-based graph clustering methods have achieved promising results. However, a deep neural network involves a large number of training parameters. Moreover, existing methods typically select the similarity metric by an ad hoc approach, which considerably affects the resulting output. In this study, we propose a principled graph learning perspective, fine-grained attributed graph clustering. Based on a shallow approach, the proposed method sufficiently exploits both node features and structure information by benefiting from graph convolution. Consequently, a fine-grained graph encoded higher-order relations is automatically learned. Comprehensive experiments on benchmark datasets demonstrate the superiority of the proposed method over state-of-the-art algorithms, including several DL methods.

In this article, we introduce a novel, general methodology, called integrate and conquer, for simultaneously accomplishing the tasks of feature extraction, manifold construction, and clustering, which is taken to be superior to building a clustering method as a single task. When the proposed novel methodology is used on two-dimensional (2D) data, it naturally induces a new clustering method highly effective on 2D data. Existing clustering algorithms usually need to convert 2D data to vectors in a preprocessing step, which, unfortunately, severely damages 2D spatial information and omits inherent structures and correlations in the original data. The induced new clustering method can overcome the matrix-vectorization-related issues to enhance the clustering performance on 2D matrices. More specifically, the proposed methodology mutually enhances three tasks of finding subspaces, learning manifolds, and constructing data representation in a seamlessly integrated fashion. When used on 2D data, we seek two projection matrices with optimal numbers of directions to project the data into low-rank, noise-mitigated, and the most expressive subspaces, in which manifolds are adaptively updated according to the projections, and new data representation is built with respect to the projected data by accounting for nonlinearity via adaptive manifolds. Consequently, the learned subspaces and manifolds are clean and intrinsic, and the new data representation is discriminative and robust. Extensive experiments have been conducted and the results confirm the effectiveness of the proposed methodology and algorithm.

Construction of a reliable similarity matrix is fundamental for graph-based clustering methods. However, most of the current work is built upon some simple manifold structure, whereas limited work has been conducted on nonlinear data sets where data reside in a union of manifolds rather than a union of subspaces. Therefore, we construct a similarity graph to capture both global and local manifold structures of the input data set. The global structure is exploited based on the self-expressive property of data in an implicit feature space using kernel methods. Since the similarity graph computation is independent of the subsequent clustering, the final results may be far from optimal. To overcome this limitation, we simultaneously learn similarity graph and clustering structure in a principled way. Experimental studies demonstrate that our proposed algorithms deliver consistently superior results to other state-of-the-art algorithms.

Matrix factorization is a useful technique for data representation in many data mining and machine learning tasks. Particularly, for data sets with all nonnegative entries, matrix factorization often requires that factor matrices be nonnegative, leading to nonnegative matrix factorization (NMF). One important application of NMF is for clustering with reduced dimensions of the data represented in the new feature space. In this paper, we propose a new graph regularized NMF method capable of feature learning and apply it to clustering. Unlike existing NMF methods that treat all features in the original feature space equally, our method distinguishes features by incorporating a feature-wise sparse approximation error matrix in the formulation. It enables important features to be more closely approximated by the factor matrices. Meanwhile, the graph of the data is constructed using cleaner features in the feature learning process, which integrates feature learning and manifold learning procedures into a unified NMF model. This distinctly differs from applying the existing graph-based NMF models after feature selection in that, when these two procedures are independently used, they often fail to align themselves toward obtaining a compact and most expressive data representation. Comprehensive experimental results demonstrate the effectiveness of the proposed method, which outperforms state-of-the-art algorithms when applied to clustering.

Matrix factorization is often used for data representation in many data mining and machine-learning problems. In particular, for a dataset without any negative entries, nonnegative matrix factorization (NMF) is often used to find a low-rank approximation by the product of two nonnegative matrices. With reduced dimensions, these matrices can be effectively used for many applications such as clustering. The existing methods of NMF are often afflicted with their sensitivity to outliers and noise in the data. To mitigate this drawback, in this paper, we consider integrating NMF into a robust principal component model, and design a robust formulation that effectively captures noise and outliers in the approximation while incorporating essential nonlinear structures. A set of comprehensive empirical evaluations in clustering applications demonstrates that the proposed method has strong robustness to gross errors and superior performance to current state-of-the-art methods.

Recommender systems play an increasingly important role in online applications to help users find what they need or prefer. Collaborative filtering algorithms that generate predictions by analyzing the user-item rating matrix perform poorly when the matrix is sparse. To alleviate this problem, this paper proposes a simple recommendation algorithm that fully exploits the similarity information among users and items and intrinsic structural information of the user-item matrix. The proposed method constructs a new representation which preserves affinity and structure information in the user-item rating matrix and then performs recommendation task. To capture proximity information about users and items, two graphs are constructed. Manifold learning idea is used to constrain the new representation to be smooth on these graphs, so as to enforce users and item proximities. Our model is formulated as a convex optimization problem, for which we need to solve the well known Sylvester equation only. We carry out extensive empirical evaluations on six benchmark datasets to show the effectiveness of this approach.

Recovering low-rank matrices is a problem common in many applications of data mining and machine learning, such as matrix completion and image denoising. Robust Principal Component Analysis (RPCA) has emerged for handling such kinds of problems; however, the existing RPCA approaches are usually computationally expensive, due to the fact that they need to obtain the singular value decomposition (SVD) of large matrices. In this paper, we propose a novel RPCA approach that eliminates the need for SVD of large matrices. Scalable algorithms are designed for several variants of our approach, which are crucial for real world applications on large scale data. Extensive experimental results confirm the effectiveness of our approach both quantitatively and visually.

Robust principal component analysis (RPCA) has been widely used for recovering low-rank matrices in many data mining and machine learning problems. It separates a data matrix into a low-rank part and a sparse part. The convex approach has been well studied in the literature. However, state-of-the-art algorithms for the convex approach usually have relatively high complexity due to the need of solving (partial) singular value decompositions of large matrices. A non-convex approach, AltProj, has also been proposed with lighter complexity and better scalability. Given the true rank r of the underlying low rank matrix, AltProj has a complexity of O(r2dn), where d × n is the size of data matrix. In this paper, we propose a novel factorization-based model of RPCA, which has a complexity of O(kdn), where k is an upper bound of the true rank. Our method does not need the precise value of the true rank. From extensive experiments, we observe that AltProj can work only when r is precisely known in advance, however, when the needed rank parameter r is specified to a value different from the true rank, AltProj cannot fully separate the two parts while our method succeeds. Even when both work, our method is about 4 times faster than AltProj. Our method can be used as a light-weight, scalable tool for RPCA in the absence of the precise value of the true rank.

The importance of accurate recommender systems has been widely recognized by academia and industry. However, the recommendation quality is still rather low. Recently, a linear sparse and low-rank representation of the user-item matrix has been applied to produce Top-N recommendations. This approach uses the nuclear norm as a convex relaxation for the rank function and has achieved better recommendation accuracy than the state-of-the-art methods. In the past several years, solving rank minimization problems by leveraging nonconvex relaxations has received increasing attention. Some empirical results demonstrate that it can provide a better approximation to original problems than convex relaxation. In this paper, we propose a novel rank approximation to enhance the performance of Top-N recommendation systems, where the approximation error is controllable. Experimental results on real data show that the proposed rank approximation improves the Top-N recommendation accuracy substantially.

Matrix rank minimization problem is in general NP-hard. The nuclear norm is used to substitute the rank function in many recent studies. Nevertheless, the nuclear norm approximation adds all singular values together and the approximation error may depend heavily on the magnitudes of singular values. This might restrict its capability in dealing with many practical problems. In this paper, an arctangent function is used as a tighter approximation to the rank function. We use it on the challenging subspace clustering problem. For this nonconvex minimization problem, we develop an effective optimization procedure based on a type of augmented Lagrange multipliers (ALM) method. Extensive experiments on face clustering and motion segmentation show that the proposed method is effective for rank approximation.

Numerous applications in data mining and machine learning require recovering a matrix of minimal rank. Robust principal component analysis (RPCA) is a general framework for handling this kind of problems. Nuclear norm based convex surrogate of the rank function in RPCA is widely investigated. Under certain assumptions, it can recover the underlying true low rank matrix with high probability. However, those assumptions may not hold in real-world applications. Since the nuclear norm approximates the rank by adding all singular values together, which is essentially a l1-norm of the singular values, the resulting approximation erroris not trivial and thus the resulting matrix estimator can be significantly biased. To seek a closer approximation and to alleviate the above-mentioned limitations of the nuclear norm, we propose a nonconvex rank approximation. This approximation to the matrix rank is tighter than the nuclear norm. To solve the associated nonconvex minimization problem, we develop an efficient augmented Lagrange multiplier based optimization algorithm. Experimental results demonstrate that our method outperforms current state-of-the-art algorithms in both accuracy and efficiency.

A number of machine learning and computer vision problems, such as matrix completion and subspace clustering, require a matrix to be of low-rank. To meet this requirement, most existing methods use the nuclear norm as a convex proxy of the rank function and minimize it. However, the nuclear norm simply adds all nonzero singular values together instead of treating them equally as the rank function does, which may not be a good rank approximation when some singular values are very large. To reduce this undesirable weighting effect, we use a log-determinant function as a non-convex rank approximation which reduces the contributions of large singular values while keeping those of small singular values close to zero. We apply the method of augmented Lagrangian multipliers to optimize this non-convex rank approximation-based objective function and obtain closed-form solutions for all subproblems of minimizing different variables alternatively. The log-determinant low-rank optimization method is used to solve subspace clustering problem, for which we construct an affinity matrix based on the angular information of the low-rank representation to enhance its separability property. Extensive experimental results on face clustering and motion segmentation data demonstrate the effectiveness of the proposed method.