Abstract The aim of this article is to develop a low-rank linear regression model to correlate a high-dimensional response matrix with a high-dimensional vector of covariates when coefficient matrices have low-rank structures. We propose a fast and efficient screening procedure based on the spectral norm of each coefficient matrix to deal with the case when the number of covariates is extremely large. We develop an efficient estimation procedure based on the trace norm regularization, which explicitly imposes the low rank structure of coefficient matrices. When both the dimension of response matrix and that of covariate vector diverge at the exponential order of the sample size, we investigate the sure independence screening property under some mild conditions. We also systematically investigate some theoretical properties of our estimation procedure including estimation consistency, rank consistency, and nonasymptotic error bound under some mild conditions. We further establish a theoretical guarantee for the overall solution of our two-step screening and estimation procedure. We examine the finite-sample performance of our screening and estimation methods using simulations and a large-scale imaging genetic dataset collected by the Philadelphia Neurodevelopmental Cohort study. Supplementary materials for this article are available online.

中文翻译：

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L2RM：高维矩阵响应的低秩线性回归模型


摘要 本文的目的是开发一种低秩线性回归模型，当系数矩阵具有低秩结构时，将高维响应矩阵与高维协变量向量相关联。我们提出了一种基于每个系数矩阵的谱范数的快速有效的筛选程序，以处理协变量数量极大的情况。我们开发了一种基于迹范数正则化的有效估计程序，它明确地施加了系数矩阵的低秩结构。当响应矩阵的维数和协变量向量的维数都以样本量的指数级发散时，我们研究了在一些温和条件下的确定的独立筛选特性。我们还系统地研究了估计过程的一些理论特性，包括估计一致性、秩一致性和一些温和条件下的非渐近误差界。我们进一步为我们的两步筛选和估计程序的整体解决方案建立了理论保证。我们使用模拟和费城神经发育队列研究收集的大规模成像遗传数据集来检查我们的筛选和估计方法的有限样本性能。本文的补充材料可在线获取。