High-dimensional datasets often contain multiple meaningful clusterings in different subspaces. For example, objects can be clustered either by color, weight, or size, revealing different interpretations of the given dataset. A variety of approaches are able to identify such non-redundant clusterings. However, most of these methods require the user to specify the expected number of subspaces and clusters for each subspace. Stating these values is a non-trivial problem and usually requires detailed knowledge of the input dataset. In this paper, we propose a framework that utilizes the Minimum Description Length Principle (MDL) to detect the number of subspaces and clusters per subspace automatically. We describe an efficient procedure that greedily searches the parameter space by splitting and merging subspaces and clusters within subspaces. Additionally, an encoding strategy is introduced that allows us to detect outliers in each subspace. Extensive experiments show that our approach is highly competitive to state-of-the-art methods.

Graph clustering aims at discovering a natural grouping of the nodes such that similar nodes are assigned to a common cluster. Many different algorithms have been proposed in the literature: for simple graphs, for graphs with attributes associated to nodes, and for graphs where edges represent different types of relations among nodes. However, complex data in many domains can be represented as both attributed and multi-relational networks. In this paper, we propose SpectralMix, a joint dimensionality reduction technique for multi-relational graphs with categorical node attributes. SpectralMix integrates all information available from the attributes, the different types of relations, and the graph structure to enable a sound interpretation of the clustering results. Moreover, it generalizes existing techniques: it reduces to spectral embedding and clustering when only applied to a single graph and to homogeneity analysis when applied to categorical data. Experiments conducted on several real-world datasets enable us to detect dependencies between graph structure and categorical attributes, moreover, they exhibit the superiority of SpectralMix over existing methods.

Multi-view data containing complementary and consensus information can facilitate representation learning by exploiting the intact integration of multi-view features. Because most objects in real world often have underlying connections, organizing multi-view data as heterogeneous graphs is beneficial to extracting latent information among different objects. Due to the powerful capability to gather information of neighborhood nodes, in this paper, we apply Graph Convolutional Network (GCN) to cope with heterogeneous-graph data originating from multi-view data, which is still under-explored in the field of GCN. In order to improve the quality of network topology and alleviate the interference of noises yielded by graph fusion, some methods undertake sorting operations before the graph convolution procedure. These GCN-based methods generally sort and select the most confident neighborhood nodes for each vertex, such as picking the top-k nodes according to pre-defined confidence values. Nonetheless, this is problematic due to the non-differentiable sorting operators and inflexible graph embedding learning, which may result in blocked gradient computations and undesired performance. To cope with these issues, we propose a joint framework dubbed Multi-view Graph Convolutional Network with Differentiable Node Selection (MGCN-DNS), which is constituted of an adaptive graph fusion layer, a graph learning module and a differentiable node selection schema. MGCN-DNS accepts multi-channel graph-structural data as inputs and aims to learn more robust graph fusion through a differentiable neural network. The effectiveness of the proposed method is verified by rigorous comparisons with considerable state-of-the-art approaches in terms of multi-view semi-supervised classification tasks.

Heterogeneous graph neural networks aim to discover discriminative node embeddings and relations from multi-relational networks.One challenge of heterogeneous graph learning is the design of learnable meta-paths, which significantly influences the quality of learned embeddings.Thus, in this paper, we propose an Attributed Multi-Order Graph Convolutional Network (AMOGCN), which automatically studies meta-paths containing multi-hop neighbors from an adaptive aggregation of multi-order adjacency matrices. The proposed model first builds different orders of adjacency matrices from manually designed node connections. After that, an intact multi-order adjacency matrix is attached from the automatic fusion of various orders of adjacency matrices. This process is supervised by the node semantic information, which is extracted from the node homophily evaluated by attributes. Eventually, we utilize a one-layer simplifying graph convolutional network with the learned multi-order adjacency matrix, which is equivalent to the cross-hop node information propagation with multi-layer graph neural networks. Substantial experiments reveal that AMOGCN gains superior semi-supervised classification performance compared with state-of-the-art competitors.

Graph Convolutional Network (GCN) with the powerful capacity to explore graph-structural data has gained noticeable success in recent years. Nonetheless, most of the existing GCN-based models suffer from the notorious over-smoothing issue, owing to which shallow networks are extensively adopted. This may be problematic for complex graph datasets because a deeper GCN should be beneficial to propagating information across remote neighbors. Recent works have devoted effort to addressing over-smoothing problems, including establishing residual connection structure or fusing predictions from multi-layer models. Because of the indistinguishable embeddings from deep layers, it is reasonable to generate more reliable predictions before conducting the combination of outputs from various layers. In light of this, we propose an Alternating Graph-regularized Neural Network (AGNN) composed of Graph Convolutional Layer (GCL) and Graph Embedding Layer (GEL). GEL is derived from the graph-regularized optimization containing Laplacian embedding term, which can alleviate the over-smoothing problem by periodic projection from the low-order feature space onto the high-order space. With more distinguishable features of distinct layers, an improved Adaboost strategy is utilized to aggregate outputs from each layer, which explores integrated embeddings of multi-hop neighbors. The proposed model is evaluated via a large number of experiments including performance comparison with some multi-layer or multi-order graph neural networks, which reveals the superior performance improvement of AGNN compared with state-of-the-art models.

Both clustering and outlier detection play an important role for meteorological measurements. We present the AWT algorithm, a clustering algorithm for time series data that also performs implicit outlier detection during the clustering. AWT integrates ideas of several well-known K-Means clustering algorithms. It chooses the number of clusters automatically based on a user-defined threshold parameter, and it can be used for heterogeneous meteorological input data as well as for data sets that exceed the available memory size. We apply AWT to crowd sourced 2-m temperature data with an hourly resolution from the city of Vienna to detect outliers and to investigate if the final clusters show general similarities and similarities with urban land-use characteristics. It is shown that both the outlier detection and the implicit mapping to land-use characteristic is possible with AWT which opens new possible fields of application, specifically in the rapidly evolving field of urban climate and urban weather.

In practical applications, multi-view data depicting objectives from assorted perspectives can facilitate the accuracy increase of learning algorithms. However, given multi-view data, there is limited work for learning discriminative node relationships and graph information simultaneously via graph convolutional network that has drawn the attention from considerable researchers in recent years. Most of existing methods only consider the weighted sum of adjacency matrices, yet a joint neural network of both feature and graph fusion is still under-explored. To cope with these issues, this paper proposes a joint deep learning framework called Learnable Graph Convolutional Network and Feature Fusion (LGCN-FF), consisting of two stages: feature fusion network and learnable graph convolutional network. The former aims to learn an underlying feature representation from heterogeneous views, while the latter explores a more discriminative graph fusion via learnable weights and a parametric activation function dubbed Differentiable Shrinkage Activation (DSA) function. The proposed LGCN-FF is validated to be superior to various state-of-the-art methods in multi-view semi-supervised classification.

The field of deep clustering combines deep learning and clustering to learn representations that improve both the learned representation and the performance of the considered clustering method. Most existing deep clustering methods are designed for a single clustering method, e.g., k-means, spectral clustering, or Gaussian mixture models, but it is well known that no clustering algorithm works best in all circumstances. Consensus clustering tries to alleviate the individual weaknesses of clustering algorithms by building a consensus between members of a clustering ensemble. Currently, there is no deep clustering method that can include multiple heterogeneous clustering algorithms in an ensemble to update representations and clusterings together. To close this gap, we introduce the idea of a consensus representation that maximizes the agreement between ensemble members. Further, we propose DECCS (Deep Embedded Clustering with Consensus representationS), a deep consensus clustering method that learns a consensus representation by enhancing the embedded space to such a degree that all ensemble members agree on a common clustering result. Our contributions are the following: (1) We introduce the idea of learning consensus representations for heterogeneous clusterings, a novel notion to approach consensus clustering. (2) We propose DECCS, the first deep clustering method that jointly improves the representation and clustering results of multiple heterogeneous clustering algorithms. (3) We show in experiments that learning a consensus representation with DECCS is outperforming several relevant baselines from deep clustering and consensus clustering. Our code can be found at https://gitlab.cs.univie.ac.at/lukas/deccs

Network-structured data becomes ubiquitous in daily life and is growing at a rapid pace. It presents great challenges to feature engineering due to the high non-linearity and sparsity of the data. The local and global structure of the real-world networks can be reflected by dynamical transfer behaviors among nodes. This paper proposes a network embedding framework to capture the transfer behaviors on structured networks via deep prediction models. We first design a degree-weight biased random walk model to capture the transfer behaviors on the network. Then a deep network embedding method is introduced to preserve the transfer possibilities among the nodes. A network structure embedding layer is added into conventional deep prediction models, including Long Short-Term Memory Network and Recurrent Neural Network, to utilize the sequence prediction ability. To keep the local network neighborhood, we further perform a Laplacian supervised space optimization on the embedding feature representations. Experimental studies are conducted on various datasets including social networks, citation networks, biomedical network, collaboration network and language network. The results show that the learned representations can be effectively used as features in a variety of tasks, such as clustering, visualization, classification, reconstruction and link prediction, and achieve promising performance compared with state-of-the-arts.