Hypergraph learning is a technique for conducting learning on a hypergraph structure. In recent years, hypergraph learning has attracted increasing attention due to its flexibility and capability in modeling complex data correlation. In this paper, we first systematically review existing literature regarding hypergraph generation, including distance-based, representation-based, attribute-based, and network-based approaches. Then, we introduce the existing learning methods on a hypergraph, including transductive hypergraph learning, inductive hypergraph learning, hypergraph structure updating, and multi-modal hypergraph learning. After that, we present a tensor-based dynamic hypergraph representation and learning framework that can effectively describe high-order correlation in a hypergraph. To study the effectiveness and efficiency of hypergraph generation and learning methods, we conduct comprehensive evaluations on several typical applications, including object and action recognition, Microblog sentiment prediction, and clustering. In addition, we contribute a hypergraph learning development toolkit called THU-HyperG.

The Kullback-Leibler divergence (KLD), which is widely used to measure the similarity between two distributions, plays an important role in many applications. In this article, we address the KLD metric-learning task, which aims at learning the best KLD-type metric from the distributions of datasets. Concretely, first, we extend the conventional KLD by introducing a linear mapping and obtain the best KLD to well express the similarity of data distributions by optimizing such a linear mapping. It improves the expressivity of data distribution, which means it makes the distributions in the same class close and those in different classes far away. Then, the KLD metric learning is modeled by a minimization problem on the manifold of all positive-definite matrices. To deal with this optimization task, we develop an intrinsic steepest descent method, which preserves the manifold structure of the metric in the iteration. Finally, we apply the proposed method along with ten popular metric-learning approaches on the tasks of 3-D object classification and document classification. The experimental results illustrate that our proposed method outperforms all other methods.

In this paper, we present a hypergraph neural networks (HGNN) framework for data representation learning, which can encode high-order data correlation in a hypergraph structure. Confronting the challenges of learning representation for complex data in real practice, we propose to incorporate such data structure in a hypergraph, which is more flexible on data modeling, especially when dealing with complex data. In this method, a hyperedge convolution operation is designed to handle the data correlation during representation learning. In this way, traditional hypergraph learning procedure can be conducted using hyperedge convolution operations efficiently. HGNN is able to learn the hidden layer representation considering the high-order data structure, which is a general framework considering the complex data correlations. We have conducted experiments on citation network classification and visual object recognition tasks and compared HGNN with graph convolutional networks and other traditional methods. Experimental results demonstrate that the proposed HGNN method outperforms recent state-of-theart methods. We can also reveal from the results that the proposed HGNN is superior when dealing with multi-modal data compared with existing methods.

At present, convolutional neural networks (CNNs) have become popular in visual classification tasks because of their superior performance. However, CNN-based methods do not consider the correlation of visual data to be classified. Recently, graph convolutional networks (GCNs) have mitigated this problem by modeling the pairwise relationship in visual data. Real-world tasks of visual classification typically must address numerous complex relationships in the data, which are not fit for the modeling of the graph structure using GCNs. Therefore, it is vital to explore the underlying correlation of visual data. Regarding this issue, we propose a framework called the hypergraph-induced convolutional network to explore the high-order correlation in visual data during deep neural networks. First, a hypergraph structure is constructed to formulate the relationship in visual data. Then, the high-order correlation is optimized by a learning process based on the constructed hypergraph. The classification tasks are performed by considering the high-order correlation in the data. Thus, the convolution of the hypergraph-induced convolutional network is based on the corresponding high-order relationship, and the optimization on the network uses each data and considers the high-order correlation of the data. To evaluate the proposed hypergraph-induced convolutional network framework, we have conducted experiments on three visual data sets: the National Taiwan University 3-D model data set, Princeton Shape Benchmark, and multiview RGB-depth object data set. The experimental results and comparison in all data sets demonstrate the effectiveness of our proposed hypergraph-induced convolutional network compared with the state-of-the-art methods.

Earth Mover's Distance (EMD), targeting at measuring the many-to-many distances, has shown its superiority and been widely applied in computer vision tasks, such as object recognition, hyperspectral image classification and gesture recognition. However, there is still little effort concentrated on optimizing the EMD metric towards better matching performance. To tackle this issue, we propose an EMD metric learning algorithm in this paper. In our method, the objective is to learn a discriminative distance metric for EMD ground distance matrix generation which can better measure the similarity between compared subjects. More specifically, given a group of labeled data from different categories, we first select a subset of training data and then optimize the metric for ground distance matrix generation. Here, both the EMD metric and the EMD flow-network are alternatively optimized until a steady EMD value can be achieved. This method is able to generate a discriminative ground distance matrix which can further improve the EMD distance measurement. We then apply our EMD metric learning method on two tasks, i.e., multi-view object classification and document classification. The experimental results have shown better performance of our proposed EMD metric learning method compared with the traditional EMD method and the state-of-the-art methods. It is noted that the proposed EMD metric learning method can be also used in other applications.

3D shape recognition has attracted much attention recently. Its recent advances advocate the usage of deep features and achieve the state-of-the-art performance. However, existing deep features for 3D shape recognition are restricted to a view-to-shape setting, which learns the shape descriptor from the view-level feature directly. Despite the exciting progress on view-based 3D shape description, the intrinsic hierarchical correlation and discriminability among views have not been well exploited, which is important for 3D shape representation. To tackle this issue, in this paper, we propose a group-view convolutional neural network (GVCNN) framework for hierarchical correlation modeling towards discriminative 3D shape description. The proposed GVCNN framework is composed of a hierarchical view-group-shape architecture, i.e., from the view level, the group level and the shape level, which are organized using a grouping strategy. Concretely, we first use an expanded CNN to extract a view level descriptor. Then, a grouping module is introduced to estimate the content discrimination of each view, based on which all views can be splitted into different groups according to their discriminative level. A group level description can be further generated by pooling from view descriptors. Finally, all group level descriptors are combined into the shape level descriptor according to their discriminative weights. Experimental results and comparison with state-of-the-art methods show that our proposed GVCNN method can achieve a significant performance gain on both the 3D shape classification and retrieval tasks.

In recent years, hypergraph modeling has shown its superiority on correlation formulation among samples and has wide applications in classification, retrieval, and other tasks. In all these works, the performance of hypergraph learning highly depends on the generated hypergraph structure. A good hypergraph structure can represent the data correlation better, and vice versa. Although hypergraph learning has attracted much attention recently, most of existing works still rely on a static hypergraph structure, and little effort concentrates on optimizing the hypergraph structure during the learning process. To tackle this problem, we propose a dynamic hypergraph structure learning method in this paper. In this method, given the originally generated hypergraph structure, the objective of our work is to simultaneously optimize the label projection matrix (the common task in hypergraph learning) and the hypergraph structure itself. More specifically, in this formulation, the label projection matrix is related to the hypergraph structure, and the hypergraph structure is associated with the data correlation from both the label space and the feature space. Here, we alternatively learn the optimal label projection matrix and the hypergraph structure, leading to a dynamic hypergraph structure during the learning process. We have applied the proposed method in the tasks of 3D shape recognition and gesture recognition. Experimental results on 4 public datasets show better performance compared with the state-of-the-art methods. We note that the proposed method can be further applied in other tasks.

The wide 3D applications have led to increasing amount of 3D object data, and thus effective 3D object classification technique has become an urgent requirement. One important and challenging task for 3D object classification is how to formulate the 3D data correlation and exploit it. Most of the previous works focus on learning optimal pairwise distance metric for object comparison, which may lose the global correlation among 3D objects. Recently, a transductive hypergraph learning has been investigated for classification, which can jointly explore the correlation among multiple objects, including both the labeled and unlabeled data. Although these methods have shown better performance, they are still limited due to 1) a considerable amount of testing data may not be available in practice and 2) the high computational cost to test new coming data. To handle this problem, considering the multi-modal representations of 3D objects in practice, we propose an inductive multi-hypergraph learning algorithm, which targets on learning an optimal projection for the multi-modal training data. In this method, all the training data are formulated in multi-hypergraph based on the features, and the inductive learning is conducted to learn the projection matrices and the optimal multi-hypergraph combination weights simultaneously. Different from the transductive learning on hypergraph, the high cost training process is off-line, and the testing process is very efficient for the inductive learning on hypergraph. We have conducted experiments on two 3D benchmarks, i.e., the NTU and the ModelNet40 data sets, and compared the proposed algorithm with the state-of-the-art methods and traditional transductive multi-hypergraph learning methods. Experimental results have demonstrated that the proposed method can achieve effective and efficient classification performance. We also note that the proposed method is a general framework and has the potential to be applied in other applications in practice.