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

2025-06-07

AI Summary

To address the challenge of jointly modeling multimodal predictors and responses in high-dimensional tensor time series, this paper proposes the first unified stochastic regression framework based on CANDECOMP/PARAFAC (CP) decomposition, supporting arbitrary combinations of vector-, matrix-, and tensor-valued inputs and outputs—including tensor autoregression as a natural special case. Methodologically, it systematically introduces CP decomposition into high-dimensional tensor regression to enable interpretable modeling of cross-modal interactions. We construct both CP low-rank and sparse CP low-rank estimators and derive tight non-asymptotic error bounds. Theoretical analysis integrates alternating minimization with regularization techniques; simulations confirm the tightness of the bounds and computational efficiency. Empirical studies on macroeconomic and air pollution data successfully uncover interpretable low-dimensional dynamic dependence structures.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Computer Vision: Multi-modal Vision

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Modeling high-dimensional tensor time series data
• Interpreting interactive roles of tensor modes
• Developing efficient estimation algorithms for tensor regression

Innovation

• CP decomposition for tensor regression
• Low-rank and sparse estimators
• Alternating minimization algorithm