基于迭代投影奇异值分解的半参数张量因子分析
Semi-parametric TEnsor Factor Analysis by Iteratively Projected Singular Value Decomposition
AI总结：
本文提出半参数张量因子分析框架STEFA及迭代投影奇异值分解算法IP-SVD，通过引入协变量提升低秩张量分解的估计精度与收敛速度，并验证其在估计和预测任务中的有效性。
摘要：
本文引入了一个通用的半参数张量因子分析（STEFA）框架，重点研究带有辅助协变量的低秩张量分解的方法论与理论。半参数张量因子分析模型通过在载荷矩阵中引入辅助协变量来扩展张量因子模型。我们提出了一种用于半参数估计的迭代投影奇异值分解（IP-SVD）算法。该算法将张量数据迭代投影到由协变量基函数张成的线性空间上，并在每个模态上对矩阵化的张量应用奇异值分解。我们建立了载荷矩阵和核心张量因子的收敛速率。理论结果仅要求次指数噪声分布，这比文献中噪声具有次高斯尾的假设更弱。与Tucker分解相比，IP-SVD能够以更快的收敛速率得到更准确的估计量。除估计外，我们还基于STEFA模型提出了几种利用新协变量进行预测的方法。在合成张量数据和真实张量数据上，我们展示了STEFA模型和IP-SVD算法在估计和预测任务上的有效性。
英文摘要：
This paper introduces a general framework of Semi-parametric TEnsor Factor Analysis (STEFA) that focuses on the methodology and theory of low-rank tensor decomposition with auxiliary covariates. Semi-parametric TEnsor Factor Analysis models extend tensor factor models by incorporating auxiliary covariates in the loading matrices. We propose an algorithm of iteratively projected singular value decomposition (IP-SVD) for the semi-parametric estimation. It iteratively projects tensor data onto the linear space spanned by the basis functions of covariates and applies singular value decomposition on matricized tensors over each mode. We establish the convergence rates of the loading matrices and the core tensor factor. The theoretical results only require a sub-exponential noise distribution, which is weaker than the assumption of sub-Gaussian tail of noise in the literature. Compared with the Tucker decomposition, IP-SVD yields more accurate estimators with a faster convergence rate. Besides estimation, we propose several prediction methods with new covariates based on the STEFA model. On both synthetic and real tensor data, we demonstrate the efficacy of the STEFA model and the IP-SVD algorithm on both the estimation and prediction tasks.
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