Zhongyuan Lyu

Ting Li, Zhongyuan Lyu, Chenyu Ren et al.

This paper develops an R package rMultiNet to analyze multilayer network data. We provide two general frameworks from recent literature, e.g. mixture multilayer stochastic block model(MMSBM) and mixture multilayer latent space model(MMLSM) to generate the multilayer network. We also provide several methods to reveal the embedding of both nodes and layers followed by further data analysis methods, such as clustering. Three real data examples are processed in the package. The source code of rMultiNet is available at https://github.com/ChenyuzZZ73/rMultiNet.

Zhongyuan Lyu, Dong Xia

This paper investigates the computational and statistical limits in clustering matrix-valued observations. We propose a low-rank mixture model (LrMM), adapted from the classical Gaussian mixture model (GMM) to treat matrix-valued observations, which assumes low-rankness for population center matrices. A computationally efficient clustering method is designed by integrating Lloyd's algorithm and low-rank approximation. Once well-initialized, the algorithm converges fast and achieves an exponential-type clustering error rate that is minimax optimal. Meanwhile, we show that a tensor-based spectral method delivers a good initial clustering. Comparable to GMM, the minimax optimal clustering error rate is decided by the separation strength, i.e., the minimal distance between population center matrices. By exploiting low-rankness, the proposed algorithm is blessed with a weaker requirement on the separation strength. Unlike GMM, however, the computational difficulty of LrMM is characterized by the signal strength, i.e., the smallest non-zero singular values of population center matrices. Evidence is provided showing that no polynomial-time algorithm is consistent if the signal strength is not strong enough, even though the separation strength is strong. Intriguing differences between estimation and clustering under LrMM are discussed. The merits of low-rank Lloyd's algorithm are confirmed by comprehensive simulation experiments. Finally, our method outperforms others in the literature on real-world datasets.

Zhongyuan Lyu, Dong Xia

Structural matrix-variate observations routinely arise in diverse fields such as multi-layer network analysis and brain image clustering. While data of this type have been extensively investigated with fruitful outcomes being delivered, the fundamental questions like its statistical optimality and computational limit are largely under-explored. In this paper, we propose a low-rank Gaussian mixture model (LrMM) assuming each matrix-valued observation has a planted low-rank structure. Minimax lower bounds for estimating the underlying low-rank matrix are established allowing a whole range of sample sizes and signal strength. Under a minimal condition on signal strength, referred to as the information-theoretical limit or statistical limit, we prove the minimax optimality of a maximum likelihood estimator which, in general, is computationally infeasible. If the signal is stronger than a certain threshold, called the computational limit, we design a computationally fast estimator based on spectral aggregation and demonstrate its minimax optimality. Moreover, when the signal strength is smaller than the computational limit, we provide evidences based on the low-degree likelihood ratio framework to claim that no polynomial-time algorithm can consistently recover the underlying low-rank matrix. Our results reveal multiple phase transitions in the minimax error rates and the statistical-to-computational gap. Numerical experiments confirm our theoretical findings. We further showcase the merit of our spectral aggregation method on the worldwide food trading dataset.

Bing-Yi Jing, Ting Li, Zhongyuan Lyu et al.

We study the problem of community detection in multi-layer networks, where pairs of nodes can be related in multiple modalities. We introduce a general framework, i.e., mixture multi-layer stochastic block model (MMSBM), which includes many earlier models as special cases. We propose a tensor-based algorithm (TWIST) to reveal both global/local memberships of nodes, and memberships of layers. We show that the TWIST procedure can accurately detect the communities with small misclassification error as the number of nodes and/or the number of layers increases. Numerical studies confirm our theoretical findings. To our best knowledge, this is the first systematic study on the mixture multi-layer networks using tensor decomposition. The method is applied to two real datasets: worldwide trading networks and malaria parasite genes networks, yielding new and interesting findings.