CP-factorization for high dimensional tensor time series and double projection iterations

2026-06-07

AI Summary

This study addresses the identification and estimation of factor loadings in high-dimensional tensor time series. Modeling the data via CP decomposition, the authors leverage temporal dependence to construct a structured matrix and obtain an initial estimate through spectral analysis, followed by a proposed double-projection iterative algorithm to refine accuracy. The method overcomes conventional assumptions requiring factors to be approximately orthogonal or independent, accommodating correlated factors, non-orthogonal loadings, sparsity, and weak factors. Theoretically, the iterative estimator achieves a faster convergence rate and possesses an asymptotically normal distribution amenable to inference. Extensive simulations and analyses of two real-world datasets demonstrate the method’s superior effectiveness and robustness across diverse scenarios.

Abstract

We adopt the canonical polyadic (CP) decomposition to model high-dimensional tensor time series. Our primary goal is to identify and estimate the factor loadings in the CP decomposition. We propose a one-pass estimation procedure through standard eigen-analysis for a matrix constructed based on the serial dependence structure of the data. The asymptotic properties of the proposed estimator are established under a general setting as long as the factor loading vectors are linearly independent, allowing the factors to be correlated and the factor loading vectors to be not nearly orthogonal. The procedure adapts to the sparsity of the factor loading vectors, accommodates weak factors, and demonstrates strong performance across a wide range of scenarios. To further reduce estimation errors, we also introduce an iterative algorithm based on a novel double projection approach. We theoretically justify the improved convergence rate of the iterative estimator, and derive the associated limiting distribution. A consistent estimator of the asymptotic variance is also provided, which plays a key role in the related inference problems. All results are validated through extensive simulations and two real data applications.

Problem

• CP-factorization
• tensor time series
• factor loadings
• high-dimensional data
• serial dependence

Innovation

• CP-factorization
• tensor time series
• double projection
• one-pass estimation
• asymptotic inference