Tensor Stochastic Regression for High-dimensional Time Series via CP Decomposition

TL;DR

This paper introduces a flexible tensor stochastic regression framework using CP decomposition for high-dimensional time series, enabling interpretable modeling of multi-mode data with theoretical guarantees and practical applications.

Contribution

It develops a unified CP-based tensor regression model that handles various data types, introduces low-rank and sparse estimators, and provides efficient algorithms with theoretical error bounds.

Findings

The proposed estimators achieve favorable non-asymptotic error bounds.

Simulation studies validate theoretical properties and computational efficiency.

Applications reveal interpretable structures and dynamic dependencies in real data.

Abstract

As tensor-valued data become increasingly common in time series analysis, there is a growing need for flexible and interpretable models that can handle high-dimensional predictors and responses across multiple modes. We propose a unified framework for high-dimensional tensor stochastic regression based on CANDECOMP/PARAFAC (CP) decomposition, which encompasses vector, matrix, and tensor responses and predictors as special cases. Tensor autoregression naturally arises as a special case within this framework. By leveraging CP decomposition, the proposed models interpret the interactive roles of any two distinct tensor modes, enabling dynamic modeling of input-output mechanisms. We develop both CP low-rank and sparse CP low-rank estimators, establish their non-asymptotic error bounds, and propose an efficient alternating minimization algorithm for estimation. Simulation studies confirm the theoretical properties and demonstrate the computational advantage. Applications to mixed-frequency macroeconomic data and spatio-temporal air pollution data reveal interpretable low-dimensional structures and meaningful dynamic dependencies.