Contextual tensor decomposition by projected alternating least squares

Tensor decomposition is among the most important tools for low-rank structure extraction in tensor data analysis. In many scenarios, contextual covariates from various domains are available, which can potentially drive the underlying structures of the observed data. The classical CANDECOMP/PARAFAC (CP) decomposition method only focuses on the relational information among objects and cannot make use of such additional information. In this article, we propose a contextual CP tensor decomposition framework which incorporates the additional covariate information to infer the intrinsic low-rank structure more accurately. Moreover, an algorithm called projected alternating least squares is proposed for parameter estimation. In particular, the projection procedure removes noise effectively from observations and improves the estimation accuracy when the dependence on covariates genuinely exists. We demonstrate the advantages of our proposal with comprehensive simulations and real data analysis on GDELT political events study. The empirical results show that our method offers an improved modeling strategy as well as efficient computation.