Scalable nonparametric multiway data analysis

Abstract Multiway data analysis deals with multiway arrays, i.e., tensors, and the goal is twofold: predicting missing entries by modeling the interactions between array elements and discovering hidden patterns, such as clusters or communities in each mode. Despite the success of existing tensor factorization approaches, they are either unable to capture nonlinear interactions, or computationally expensive to handle massive data. In addition, most of the existing methods lack a principled way to discover latent clusters, which is important for better understanding of the data. To address these issues, we propose a scalable nonparametric tensor decomposition model. It employs Dirichlet process mixture (DPM) prior to model the latent clusters; it uses local Gaussian processes (GPs) to capture nonlinear relationships and to improve scalability. An efficient online variational Bayes Expectation-Maximization algorithm is proposed to learn the model. Experiments on both synthetic and real-world data show that the proposed model is able to discover latent clusters with higher prediction accuracy than competitive methods. Furthermore, the proposed model obtains significantly better predictive performance than the state-of-the-art large scale tensor decomposition algorithm, GigaTensor, on two large datasets with billions of entries

Zero-Truncated Poisson Tensor Factorization for Massive Binary Tensors

We present a scalable Bayesian model for low-rank factorization of massive tensors with binary observations. The proposed model has the following key properties: (1) in contrast to the models based on the logistic or probit likelihood, using a zero-truncated Poisson likelihood for binary data allows our model to scale up in the number of \emph{ones} in the tensor, which is especially appealing for massive but sparse binary tensors; (2) side-information in form of binary pairwise relationships (e.g., an adjacency network) between objects in any tensor mode can also be leveraged, which can be especially useful in "cold-start" settings; and (3) the model admits simple Bayesian inference via batch, as well as \emph{online} MCMC; the latter allows scaling up even for \emph{dense} binary data (i.e., when the number of ones in the tensor/network is also massive). In addition, non-negative factor matrices in our model provide easy interpretability, and the tensor rank can be inferred from the data. We evaluate our model on several large-scale real-world binary tensors, achieving excellent computational scalability, and also demonstrate its usefulness in leveraging side-information provided in form of mode-network(s).Comment: UAI (Uncertainty in Artificial Intelligence) 201

Scalable Bayesian Non-Negative Tensor Factorization for Massive Count Data

We present a Bayesian non-negative tensor factorization model for count-valued tensor data, and develop scalable inference algorithms (both batch and online) for dealing with massive tensors. Our generative model can handle overdispersed counts as well as infer the rank of the decomposition. Moreover, leveraging a reparameterization of the Poisson distribution as a multinomial facilitates conjugacy in the model and enables simple and efficient Gibbs sampling and variational Bayes (VB) inference updates, with a computational cost that only depends on the number of nonzeros in the tensor. The model also provides a nice interpretability for the factors; in our model, each factor corresponds to a "topic". We develop a set of online inference algorithms that allow further scaling up the model to massive tensors, for which batch inference methods may be infeasible. We apply our framework on diverse real-world applications, such as \emph{multiway} topic modeling on a scientific publications database, analyzing a political science data set, and analyzing a massive household transactions data set.Comment: ECML PKDD 201

Dynamic Tensor Decomposition via Neural Diffusion-Reaction Processes

Tensor decomposition is an important tool for multiway data analysis. In practice, the data is often sparse yet associated with rich temporal information. Existing methods, however, often under-use the time information and ignore the structural knowledge within the sparsely observed tensor entries. To overcome these limitations and to better capture the underlying temporal structure, we propose Dynamic EMbedIngs fOr dynamic Tensor dEcomposition (DEMOTE). We develop a neural diffusion-reaction process to estimate dynamic embeddings for the entities in each tensor mode. Specifically, based on the observed tensor entries, we build a multi-partite graph to encode the correlation between the entities. We construct a graph diffusion process to co-evolve the embedding trajectories of the correlated entities and use a neural network to construct a reaction process for each individual entity. In this way, our model can capture both the commonalities and personalities during the evolution of the embeddings for different entities. We then use a neural network to model the entry value as a nonlinear function of the embedding trajectories. For model estimation, we combine ODE solvers to develop a stochastic mini-batch learning algorithm. We propose a stratified sampling method to balance the cost of processing each mini-batch so as to improve the overall efficiency. We show the advantage of our approach in both simulation study and real-world applications. The code is available at https://github.com/wzhut/Dynamic-Tensor-Decomposition-via-Neural-Diffusion-Reaction-Processes

Bayesian Methods in Tensor Analysis

Tensors, also known as multidimensional arrays, are useful data structures in machine learning and statistics. In recent years, Bayesian methods have emerged as a popular direction for analyzing tensor-valued data since they provide a convenient way to introduce sparsity into the model and conduct uncertainty quantification. In this article, we provide an overview of frequentist and Bayesian methods for solving tensor completion and regression problems, with a focus on Bayesian methods. We review common Bayesian tensor approaches including model formulation, prior assignment, posterior computation, and theoretical properties. We also discuss potential future directions in this field.Comment: 32 pages, 8 figures, 2 table

Bayesian Robust Tensor Factorization for Incomplete Multiway Data

We propose a generative model for robust tensor factorization in the presence of both missing data and outliers. The objective is to explicitly infer the underlying low-CP-rank tensor capturing the global information and a sparse tensor capturing the local information (also considered as outliers), thus providing the robust predictive distribution over missing entries. The low-CP-rank tensor is modeled by multilinear interactions between multiple latent factors on which the column sparsity is enforced by a hierarchical prior, while the sparse tensor is modeled by a hierarchical view of Student-$t$ distribution that associates an individual hyperparameter with each element independently. For model learning, we develop an efficient closed-form variational inference under a fully Bayesian treatment, which can effectively prevent the overfitting problem and scales linearly with data size. In contrast to existing related works, our method can perform model selection automatically and implicitly without need of tuning parameters. More specifically, it can discover the groundtruth of CP rank and automatically adapt the sparsity inducing priors to various types of outliers. In addition, the tradeoff between the low-rank approximation and the sparse representation can be optimized in the sense of maximum model evidence. The extensive experiments and comparisons with many state-of-the-art algorithms on both synthetic and real-world datasets demonstrate the superiorities of our method from several perspectives.Comment: in IEEE Transactions on Neural Networks and Learning Systems, 201