Graph neural networks (GNNs) extends the functionality of traditional neural networks to graph-structured data. Similar to CNNs, an optimized design of graph convolution and pooling is key to success. Borrowing ideas from physics, we propose a path i
ntegral based graph neural networks (PAN) for classification and regression tasks on graphs. Specifically, we consider a convolution operation that involves every path linking the message sender and receiver with learnable weights depending on the path length, which corresponds to the maximal entropy random walk. It generalizes the graph Laplacian to a new transition matrix we call maximal entropy transition (MET) matrix derived from a path integral formalism. Importantly, the diagonal entries of the MET matrix are directly related to the subgraph centrality, thus providing a natural and adaptive pooling mechanism. PAN provides a versatile framework that can be tailored for different graph data with varying sizes and structures. We can view most existing GNN architectures as special cases of PAN. Experimental results show that PAN achieves state-of-the-art performance on various graph classification/regression tasks, including a new benchmark dataset from statistical mechanics we propose to boost applications of GNN in physical sciences.
Nonnegative Matrix Factorization (NMF) aims to factorize a matrix into two optimized nonnegative matrices appropriate for the intended applications. The method has been widely used for unsupervised learning tasks, including recommender systems (ratin
g matrix of users by items) and document clustering (weighting matrix of papers by keywords). However, traditional NMF methods typically assume the number of latent factors (i.e., dimensionality of the loading matrices) to be fixed. This assumption makes them inflexible for many applications. In this paper, we propose a nonparametric NMF framework to mitigate this issue by using dependent Indian Buffet Processes (dIBP). In a nutshell, we apply a correlation function for the generation of two stick weights associated with each pair of columns of loading matrices, while still maintaining their respective marginal distribution specified by IBP. As a consequence, the generation of two loading matrices will be column-wise (indirectly) correlated. Under this same framework, two classes of correlation function are proposed (1) using Bivariate beta distribution and (2) using Copula function. Both methods allow us to adopt our work for various applications by flexibly choosing an appropriate parameter settings. Compared with the other state-of-the art approaches in this area, such as using Gaussian Process (GP)-based dIBP, our work is seen to be much more flexible in terms of allowing the two corresponding binary matrix columns to have greater variations in their non-zero entries. Our experiments on the real-world and synthetic datasets show that three proposed models perform well on the document clustering task comparing standard NMF without predefining the dimension for the factor matrices, and the Bivariate beta distribution-based and Copula-based models have better flexibility than the GP-based model.
Nonnegative Matrix Factorization (NMF) aims to factorize a matrix into two optimized nonnegative matrices and has been widely used for unsupervised learning tasks such as product recommendation based on a rating matrix. However, although networks bet
ween nodes with the same nature exist, standard NMF overlooks them, e.g., the social network between users. This problem leads to comparatively low recommendation accuracy because these networks are also reflections of the nature of the nodes, such as the preferences of users in a social network. Also, social networks, as complex networks, have many different structures. Each structure is a composition of links between nodes and reflects the nature of nodes, so retaining the different network structures will lead to differences in recommendation performance. To investigate the impact of these network structures on the factorization, this paper proposes four multi-level network factorization algorithms based on the standard NMF, which integrates the vertical network (e.g., rating matrix) with the structures of horizontal network (e.g., user social network). These algorithms are carefully designed with corresponding convergence proofs to retain four desired network structures. Experiments on synthetic data show that the proposed algorithms are able to preserve the desired network structures as designed. Experiments on real-world data show that considering the horizontal networks improves the accuracy of document clustering and recommendation with standard NMF, and various structures show their differences in performance on these two tasks. These results can be directly used in document clustering and recommendation systems.
Traditional Relational Topic Models provide a way to discover the hidden topics from a document network. Many theoretical and practical tasks, such as dimensional reduction, document clustering, link prediction, benefit from this revealed knowledge.
However, existing relational topic models are based on an assumption that the number of hidden topics is known in advance, and this is impractical in many real-world applications. Therefore, in order to relax this assumption, we propose a nonparametric relational topic model in this paper. Instead of using fixed-dimensional probability distributions in its generative model, we use stochastic processes. Specifically, a gamma process is assigned to each document, which represents the topic interest of this document. Although this method provides an elegant solution, it brings additional challenges when mathematically modeling the inherent network structure of typical document network, i.e., two spatially closer documents tend to have more similar topics. Furthermore, we require that the topics are shared by all the documents. In order to resolve these challenges, we use a subsampling strategy to assign each document a different gamma process from the global gamma process, and the subsampling probabilities of documents are assigned with a Markov Random Field constraint that inherits the document network structure. Through the designed posterior inference algorithm, we can discover the hidden topics and its number simultaneously. Experimental results on both synthetic and real-world network datasets demonstrate the capabilities of learning the hidden topics and, more importantly, the number of topics.
Incorporating the side information of text corpus, i.e., authors, time stamps, and emotional tags, into the traditional text mining models has gained significant interests in the area of information retrieval, statistical natural language processing,
and machine learning. One branch of these works is the so-called Author Topic Model (ATM), which incorporates the authorss interests as side information into the classical topic model. However, the existing ATM needs to predefine the number of topics, which is difficult and inappropriate in many real-world settings. In this paper, we propose an Infinite Author Topic (IAT) model to resolve this issue. Instead of assigning a discrete probability on fixed number of topics, we use a stochastic process to determine the number of topics from the data itself. To be specific, we extend a gamma-negative binomial process to three levels in order to capture the author-document-keyword hierarchical structure. Furthermore, each document is assigned a mixed gamma process that accounts for the multi-authors contribution towards this document. An efficient Gibbs sampling inference algorithm with each conditional distribution being closed-form is developed for the IAT model. Experiments on several real-world datasets show the capabilities of our IAT model to learn the hidden topics, authors interests on these topics and the number of topics simultaneously.
