Semiparametric Tensor Factor Analysis by Iteratively Projected SVD

This paper introduces a general framework of Semiparametric TEnsor FActor analysis (STEFA) that focuses on the methodology and theory of low-rank tensor decomposition with auxiliary covariates. STEFA models extend tensor factor models by incorporating instrumental covariates in the loading matrices. We propose an algorithm of Iteratively Projected SVD (IP-SVD) for the semiparametric estimations. It iteratively projects tensor data onto the linear space spanned by covariates and applies SVD on matricized tensors over each mode. We establish the convergence rates of the loading matrices and the core tensor factor. Compared with the Tucker decomposition, IP-SVD yields more accurate estimates with a faster convergence rate. Besides estimation, we show several prediction methods with newly observed covariates based on the STEFA model. On both real and synthetic tensor data, we demonstrate the efficacy of the STEFA model and the IP-SVD algorithm on both the estimation and prediction tasks.