Traditional shortest-path graph kernels generate for each graph a histogram-like feature map, whose elements represent the number of occurrences of non-isomorphic shortest paths in this graph. The histogram-like feature map does not contain the distributions of the shortest paths within and across graphs, causing inaccurate graph similarities. To this end, we propose a novel graph kernel called the Distributional Shortest-Path (DSP) graph kernel to embrace both types of distribution information. Since the distribution of substructures (e.g., the shortest paths) follows a power law like that of words in natural language, we utilize neural language models to learn each node's distributional shortest-path feature map, encompassing the distributions and dependencies of the shortest paths in each graph. Moreover, we design the Partition Kernel (PK) to capture the dataset-wide distribution information of the shortest paths. PK projects similar (i.e., belonging to the same partition) distributional shortest-path node feature maps to the same point in the Reproducing Kernel Hilbert Space. Finally, Kernel Mean Embedding (KME) is applied to compute graph feature maps and efficiently construct the DSP graph kernel. Empirical experiments demonstrate that DSP outperforms state-of-the-art graph kernels on most benchmark datasets.

R-convolution graph kernels are positive-semidefinite functions that decompose graphs into substructures and compare them. One problem in the effective implementation of this idea is that the substructures are not independent, which leads to high-dimensional feature space. In addition, graph kernels cannot capture the high-order complex interactions between vertices. To mitigate these two problems, we propose a framework called DeepMap to learn deep representations for graph feature maps used in graph kernels. The learned deep representation for a graph is a dense and low-dimensional vector that captures complex high-order interactions in a vertex neighborhood. We empirically validate DeepMap on various graph classification benchmarks and demonstrate that it achieves state-of-the-art performance.

In contrast to global graph clustering, local graph clustering aims to find only one cluster that is concentrating on the given seed vertex (and also on the designated attributes for attributed graphs). Currently, very few methods can deal with this kind of task. To this end, we propose two quality measures for a local graph cluster: Graph Unimodality (GU) and Attribute Unimodality (AU). They measure the homogeneity/unimodality of the graph structure and the subspace that is composed of the designated attributes, respectively. We call their linear combination Compactness. Further, we propose LOCLU to optimize the Compactness score in order to find a good local graph cluster. The local graph cluster detected by LOCLU concentrates on the region of interest, provides efficient information flow in the graph, and exhibits a unimodal data distribution in the subspace of the designated attributes.

Graph-structured data arise in many scenarios. A fundamental problem is to quantify the similarities of graphs for tasks such as classification. R-convolution graph kernels are positive-semidefinite functions that decompose graphs into substructures and compare them. One problem in the effective implementation of this idea is that the substructures are not independent, which leads to high-dimensional feature space. In addition, graph kernels cannot capture the high-order complex interactions between vertices. To mitigate these two problems, we propose a framework calledDeepMapto learn deep representations for graph feature maps. The learned deep representation for a graph is a dense and low-dimensional vector that captures complex high-order interactions in a vertex neighborhood.DeepMapextends Convolutional Neural Networks (CNNs) to arbitrary graphs by generating aligned vertex sequences and building the receptive field for each vertex. We empirically validateDeepMapon various graph classification benchmarks and demonstrate that it achieves state-of-the-art performance.

Graph clustering has been studied extensively on both plain graphs and attributed graphs. However, all these methods need to partition the whole graph to find cluster structures. Sometimes, based on domain knowledge, people may have information about a specific target region in the graph and only want to find a single cluster concentrated on this local region. Such a task is called local clustering. In contrast to global clustering, local clustering aims to find only one cluster that is concentrating on the given seed vertex (and also on the designated attributes for attributed graphs). Currently, very few methods can deal with this kind of task. To this end, we propose two quality measures for a local cluster: Graph Unimodality (GU) and Attribute Unimodality (AU). The former measures the homogeneity of the graph structure while the latter measures the homogeneity of the subspace that is composed of the designated attributes. We call their linear combination asCompactness. Further, we propose LOCLU to optimize theCompactnessscore. The local cluster detected by LOCLU concentrates on the region of interest, provides efficient information flow in the graph and exhibits a unimodal data distribution in the subspace of the designated attributes.

Graph-structured data arise ubiquitously in many application domains. A fundamental problem is to quantify their similarities. Graph kernels are often used for this purpose, which decompose graphs into substructures and compare these substructures. However, most of the existing graph kernels do not have the property of scale-adaptivity, i.e., they cannot compare graphs at multiple levels of granularities. Many real-world graphs such as molecules exhibit structure at varying levels of granularities. To tackle this problem, we propose a new graph kernel called Tree++ in this paper. At the heart of Tree++ is a graph kernel called the path-pattern graph kernel. The path-pattern graph kernel first builds a truncated BFS tree rooted at each vertex and then uses paths from the root to every vertex in the truncated BFS tree as features to represent graphs. The path-pattern graph kernel can only capture graph similarity at fine granularities. In order to capture graph similarity at coarse granularities, we incorporate a new concept called super path into it. The super path contains truncated BFS trees rooted at the vertices in a path. Our evaluation on a variety of real-world graphs demonstrates that Tree++ achieves the best classification accuracy compared with previous graph kernels.

A huge object collection in high-dimensional space can often be clustered in more than one way, for instance, objects could be clustered by their shape or alternatively by their color. Each grouping represents a different view of the dataset. The new research field of non-redundant clustering addresses this class of problems. In this article, we follow the approach that different, non-redundant k -means-like clusterings may exist in different, arbitrarily oriented subspaces of the high-dimensional space. We assume that these subspaces (and optionally a further noise space without any cluster structure) are orthogonal to each other. This assumption enables a particularly rigorous mathematical treatment of the non-redundant clustering problem and thus a particularly efficient algorithm, which we call N r -K means (for non-redundant k -means). The superiority of our algorithm is demonstrated both theoretically, as well as in extensive experiments. Further, we propose an extension of N r -K means that harnesses Hartigan’s dip test to identify the number of clusters for each subspace automatically.

A huge object collection in high-dimensional space can often be clustered in more than one way, for instance, objects could be clustered by their shape or alternatively by their color. Each grouping represents a different view of the data set. The new research field of non-redundant clustering addresses this class of problems. In this paper, we follow the approach that different, non-redundant k-means-like clusterings may exist in different, arbitrarily oriented subspaces of the high-dimensional space. We assume that these subspaces (and optionally a further noise space without any cluster structure) are orthogonal to each other. This assumption enables a particularly rigorous mathematical treatment of the non-redundant clustering problem and thus a particularly efficient algorithm, which we call Nr-Kmeans (for non-redundant k-means). The superiority of our algorithm is demonstrated both theoretically, as well as in extensive experiments.

Is there an optimal dimensionality reduction for k-means, revealing the prominent cluster structure hidden in the data? We propose SUBKMEANS, which extends the classic k-means algorithm. The goal of this algorithm is twofold: find a sufficient k-means-style clustering partition and transform the clusters onto a common subspace, which is optimal for the cluster structure. Our solution is able to pursue these two goals simultaneously. The dimensionality of this subspace is found automatically and therefore the algorithm comes without the burden of additional parameters. At the same time this subspace helps to mitigate the curse of dimensionality. The SUBKMEANS optimization algorithm is intriguingly simple and efficient. It is easy to implement and can readily be adopted to the current situation. Furthermore, it is compatible to many existing extensions and improvements of k-means.

Networks such as social networks, citation networks, protein-protein interaction networks, etc., are prevalent in real world. However, only very few vertices have labels compared to large amounts of unlabeled vertices. For example, in social networks, not every user provides his/her profile information such as the personal interests which are relevant for targeted advertising. Can we leverage the limited user information and friendship network wisely to infer the labels of unlabeled users?

Data can encapsulate different object groupings in subspaces of arbitrary dimension and orientation. Finding such subspaces and the groupings within them is the goal of generalized subspace clustering. In this work we present a generalized subspace clustering technique capable of finding multiple non-redundant clusterings in arbitrarily-oriented subspaces. We use Independent Subspace Analysis (ISA) to find the subspace collection that minimizes the statistical dependency (redundancy) between clusterings. We then cluster in the arbitrarily-oriented subspaces identified by ISA. Our algorithm ISAAC (Independent Subspace Analysis and Clustering) uses the Minimum Description Length principle to automatically choose parameters that are otherwise difficult to set. We comprehensively demonstrate the effectiveness of our approach on synthetic and real-world data.

Multi-scale data which contains structures at different scales of size and density is a big challenge for spectral clustering. Even given a suitable locally scaled affinity matrix, the first k eigenvectors of such a matrix still cannot separate clusters well. Thus, in this paper, we exploit the fusion of the cluster-separation information from all eigenvectors to achieve a better clustering result. Our method FUll Spectral ClustEring (FUSE) is based on Power Iteration (PI) and Independent Component Analysis (ICA). PI is used to fuse all eigenvectors to one pseudo-eigenvector which inherits all the cluster-separation information. To conquer the cluster-collision problem, we utilize PI to generate p (p > k) pseudo-eigenvectors. Since these pseudo-eigenvectors are redundant and the cluster-separation information is contaminated with noise, ICA is adopted to rotate the pseudo-eigenvectors to make them pairwise statistically independent. To let ICA overcome local optima and speed up the search process, we develop a self-adaptive and self-learning greedy search method. Finally, we select k rotated pseudo-eigenvectors (independent components) which have more cluster-separation information measured by kurtosis for clustering. Various synthetic and real-world data verifies the effectiveness and efficiency of our FUSE method.