Network-based Neighborhood regression

Given the ubiquity of modularity in biological systems, module-level regulation analysis is vital for understanding biological systems across various levels and their dynamics. Current statistical analysis on biological modules predominantly focuses on either detecting the functional modules in biological networks or sub-group regression on the biological features without using the network data. This paper proposes a novel network-based neighborhood regression framework whose regression functions depend on both the global community-level information and local connectivity structures among entities. An efficient community-wise least square optimization approach is developed to uncover the strength of regulation among the network modules while enabling asymptotic inference. With random graph theory, we derive non-asymptotic estimation error bounds for the proposed estimator, achieving exact minimax optimality. Unlike the root-n consistency typical in canonical linear regression, our model exhibits linear consistency in the number of nodes n, highlighting the advantage of incorporating neighborhood information. The effectiveness of the proposed framework is further supported by extensive numerical experiments. Application to whole-exome sequencing and RNA-sequencing Autism datasets demonstrates the usage of the proposed method in identifying the association between the gene modules of genetic variations and the gene modules of genomic differential expressions.

Tensorized Hypergraph Neural Networks

Hypergraph neural networks (HGNN) have recently become attractive and received significant attention due to their excellent performance in various domains. However, most existing HGNNs rely on first-order approximations of hypergraph connectivity patterns, which ignores important high-order information. To address this issue, we propose a novel adjacency-tensor-based Tensorized Hypergraph Neural Network (THNN). THNN is a faithful hypergraph modeling framework through high-order outer product feature message passing and is a natural tensor extension of the adjacency-matrix-based graph neural networks. The proposed THNN is equivalent to an high-order polynomial regression scheme, which enable THNN with the ability to efficiently extract high-order information from uniform hypergraphs. Moreover, in consideration of the exponential complexity of directly processing high-order outer product features, we propose using a partially symmetric CP decomposition approach to reduce model complexity to a linear degree. Additionally, we propose two simple yet effective extensions of our method for non-uniform hypergraphs commonly found in real-world applications. Results from experiments on two widely used hypergraph datasets for 3-D visual object classification show the promising performance of the proposed THNN.

Transfer learning for tensor Gaussian graphical models

Tensor Gaussian graphical models (GGMs), interpreting conditional independence structures within tensor data, have important applications in numerous areas. Yet, the available tensor data in one single study is often limited due to high acquisition costs. Although relevant studies can provide additional data, it remains an open question how to pool such heterogeneous data. In this paper, we propose a transfer learning framework for tensor GGMs, which takes full advantage of informative auxiliary domains even when non-informative auxiliary domains are present, benefiting from the carefully designed data-adaptive weights. Our theoretical analysis shows substantial improvement of estimation errors and variable selection consistency on the target domain under much relaxed conditions, by leveraging information from auxiliary domains. Extensive numerical experiments are conducted on both synthetic tensor graphs and a brain functional connectivity network data, which demonstrates the satisfactory performance of the proposed method.

Community Detection in General Hypergraph via Graph Embedding

Network data has attracted tremendous attention in recent years, and most conventional networks focus on pairwise interactions between two vertices. However, real-life network data may display more complex structures, and multi-way interactions among vertices arise naturally. In this article, we propose a novel method for detecting community structure in general hypergraph networks, uniform or non-uniform. The proposed method introduces a null vertex to augment a non-uniform hypergraph into a uniform multi-hypergraph, and then embeds the multi-hypergraph in a low-dimensional vector space such that vertices within the same community are close to each other. The resultant optimization task can be efficiently tackled by an alternative updating scheme. The asymptotic consistencies of the proposed method are established in terms of both community detection and hypergraph estimation, which are also supported by numerical experiments on some synthetic and real-life hypergraph networks.