Semi-parametric TEnsor Factor Analysis by Iteratively Projected Singular Value Decomposition

TL;DR

This paper proposes a semi-parametric tensor factor analysis framework incorporating covariates, with an iterative SVD algorithm that offers improved accuracy and convergence, supported by theoretical guarantees and empirical validation.

Contribution

It introduces the STEFA model with an IP-SVD algorithm for semi-parametric tensor decomposition, extending tensor factor models with covariates and weaker noise assumptions.

Findings

IP-SVD achieves faster convergence than Tucker decomposition.

STEFA provides more accurate estimators in tensor analysis.

Theoretical convergence rates are established under sub-exponential noise.

Abstract

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.