Abstract

Higher-order tensors have received increased attention across science and engineering. While most tensor decomposition methods are developed for a single tensor observation, scientific studies often collect side information, in the form of node features and interactions thereof, together with the tensor data. Such data problems are common in neuroimaging, network analysis, and spatial-temporal modeling. Identifying the relationship between a high-dimensional tensor and side information is important yet challenging. Here, we develop a tensor decomposition method that incorporates multiple feature matrices as side information. Unlike unsupervised tensor decomposition, our supervised decomposition captures the effective dimension reduction of the data tensor confined to feature space of interest. An efficient alternating optimization algorithm with provable spectral initialization is further developed. Our proposal handles a broad range of data types, including continuous, count, and binary observations. We apply the method to diffusion tensor imaging data from human connectome project and multi-relational political network data. We identify the key global connectivity pattern and pinpoint the local regions that are associated with available features. The package and data used are available at https://CRAN.R-project.org/package=tensorregress. Supplementary materials for this article are available online.

摘要

高阶张量在科学和工程领域受到越来越多的关注。虽然大多数张量分解方法都是针对单个张量观察而开发的，但科学研究通常会以节点特征及其交互的形式收集辅助信息以及张量数据。此类数据问题在神经成像、网络分析和时空建模中很常见。识别高维张量和边信息之间的关系很重要，但也具有挑战性。在这里，我们开发了一种张量分解方法，它结合了多个特征矩阵作为辅助信息。与无监督张量分解不同，我们的监督分解捕获了限制在感兴趣特征空间的数据张量的有效降维。进一步开发了一种具有可证明谱初始化的有效交替优化算法。我们的提议处理广泛的数据类型，包括连续、计数和二进制观察。我们将该方法应用于来自人类连接组项目和多关系政治网络数据的扩散张量成像数据。我们确定了关键的全球连通性模式并确定了与可用特征相关的局部区域。使用的包和数据可在 https://CRAN.R-project.org/package=tensorregress 获得。本文的补充材料可在线获取。我们将该方法应用于来自人类连接组项目和多关系政治网络数据的扩散张量成像数据。我们确定了关键的全球连通性模式并确定了与可用特征相关的局部区域。使用的包和数据可在 https://CRAN.R-project.org/package=tensorregress 获得。本文的补充材料可在线获取。我们将该方法应用于来自人类连接组项目和多关系政治网络数据的扩散张量成像数据。我们确定了关键的全球连通性模式并确定了与可用特征相关的局部区域。使用的包和数据可在 https://CRAN.R-project.org/package=tensorregress 获得。本文的补充材料可在线获取。