Generalized Connectivity Matrix Response Regression with Applications in Brain Connectivity Studies

Multiple-subject network data are fast emerging in recent years, where a separate connectivity matrix is measured over a common set of nodes for each individual subject, along with subject covariates information. In this article, we propose a new generalized matrix response regression model, where the observed network is treated as a matrix-valued response and the subject covariates as predictors. The new model characterizes the population-level connectivity pattern through a low-rank intercept matrix, and the effect of subject covariates through a sparse slope tensor. We develop an efficient alternating gradient descent algorithm for parameter estimation, and establish the non-asymptotic error bound for the actual estimator from the algorithm, which quantifies the interplay between the computational and statistical errors. We further show the strong consistency for graph community recovery, as well as the edge selection consistency. We demonstrate the efficacy of our method through simulations and two brain connectivity studies.

Optimal High-order Tensor SVD via Tensor-Train Orthogonal Iteration

This paper studies a general framework for high-order tensor SVD. We propose a new computationally efficient algorithm, tensor-train orthogonal iteration (TTOI), that aims to estimate the low tensor-train rank structure from the noisy high-order tensor observation. The proposed TTOI consists of initialization via TT-SVD [1] and new iterative backward/forward updates. We develop the general upper bound on estimation error for TTOI with the support of several new representation lemmas on tensor matricizations. By developing a matching information-theoretic lower bound, we also prove that TTOI achieves the minimax optimality under the spiked tensor model. The merits of the proposed TTOI are illustrated through applications to estimation and dimension reduction of high-order Markov processes, numerical studies, and a real data example on New York City taxi travel records. The software of the proposed algorithm is available online (https://github.com/Lili-Zheng-stat/TTOI).

Generalized Liquid Association Analysis for Multimodal Data Integration

Multimodal data are now prevailing in scientific research. One of the central questions in multimodal integrative analysis is to understand how two data modalities associate and interact with each other given another modality or demographic variables. The problem can be formulated as studying the associations among three sets of random variables, a question that has received relatively less attention in the literature. In this article, we propose a novel generalized liquid association analysis method, which offers a new and unique angle to this important class of problems of studying three-way associations. We extend the notion of liquid association of Li (2002) from the univariate setting to the sparse, multivariate, and high-dimensional setting. We establish a population dimension reduction model, transform the problem to sparse Tucker decomposition of a three-way tensor, and develop a higher-order orthogonal iteration algorithm for parameter estimation. We derive the non-asymptotic error bound and asymptotic consistency of the proposed estimator, while allowing the variable dimensions to be larger than and diverge with the sample size. We demonstrate the efficacy of the method through both simulations and a multimodal neuroimaging application for Alzheimer's disease research.

Multivariate Temporal Point Process Regression

Point process modeling is gaining increasing attention, as point process type data are emerging in a large variety of scientific applications. In this article, motivated by a neuronal spike trains study, we propose a novel point process regression model, where both the response and the predictor can be a high-dimensional point process. We model the predictor effects through the conditional intensities using a set of basis transferring functions in a convolutional fashion. We organize the corresponding transferring coefficients in the form of a three-way tensor, then impose the low-rank, sparsity, and subgroup structures on this coefficient tensor. These structures help reduce the dimensionality, integrate information across different individual processes, and facilitate the interpretation. We develop a highly scalable optimization algorithm for parameter estimation. We derive the large sample error bound for the recovered coefficient tensor, and establish the subgroup identification consistency, while allowing the dimension of the multivariate point process to diverge. We demonstrate the efficacy of our method through both simulations and a cross-area neuronal spike trains analysis in a sensory cortex study.

Tensor envelope mixture model for simultaneous clustering and multiway dimension reduction

In the form of multidimensional arrays, tensor data have become increasingly prevalent in modern scientific studies and biomedical applications such as computational biology, brain imaging analysis, and process monitoring system. These data are intrinsically heterogeneous with complex dependencies and structure. Therefore, ad-hoc dimension reduction methods on tensor data may lack statistical efficiency and can obscure essential findings. Model-based clustering is a cornerstone of multivariate statistics and unsupervised learning; however, existing methods and algorithms are not designed for tensor-variate samples. In this article, we propose a tensor envelope mixture model (TEMM) for simultaneous clustering and multiway dimension reduction of tensor data. TEMM incorporates tensor-structure-preserving dimension reduction into mixture modeling and drastically reduces the number of free parameters and estimative variability. An expectation-maximization-type algorithm is developed to obtain likelihood-based estimators of the cluster means and covariances, which are jointly parameterized and constrained onto a series of lower dimensional subspaces known as the tensor envelopes. We demonstrate the encouraging empirical performance of the proposed method in extensive simulation studies and a real data application in comparison with existing vector and tensor clustering methods.

A Low-Rank Tensor Factorization Using Implicit Similarity in Trust Relationships

Low-rank tensor factorization can not only mine the implicit relationships between data but also fill in the missing data when working with complex data. Compared with the traditional collaborative filtering (CF) algorithm, the changes are essentially proposed, from traditional matrix analysis to three-dimensional spatial analysis. Based on low-rank tensor factorization, this paper proposes a recommendation model that comprehensively considers local information and global information, in other words, combining the similarity between trust users and low-rank tensor factorization. First, the similarity between trusted users is measured to capture local information between users by trusting similar preferences of users when selecting items. Then, the users’ similarity is integrated into the tensor, and the low-rank tensor factorization is used to better maintain and describe the internal structure of the data to obtain global information. Furthermore, based on the idea of the alternating least squares method, the conjugate gradient (CG) optimization algorithm for the model of this paper is designed. The local and global information is used to generate the optimal expected result in an iterative process. Finally, we conducted a large number of comparative experiments on the Ciao dataset and the FilmTrust dataset. Experimental results show that the algorithm has less precision loss under the data set with lower density. Thus, not only can a perfect compromise between accuracy and coverage be achieved, but also the computational complexity can be reduced to meet the need for real-time results.