[Paper Review] Modelling matrix time series via a tensor CP-decomposition

Jinyuan Chang, Jing He+2 more|arXiv (Cornell University)|Dec 31, 2021

Tensor decomposition and applications4 citations

TL;DR

This paper proposes a novel one-pass estimation procedure for tensor CP-decomposition of matrix time series, leveraging generalized eigenanalysis to exploit serial dependence and overcome limitations of iterative methods like ALS. The method achieves consistent estimation of component vectors with established convergence rates and demonstrates superior finite-sample performance through a refined projection technique that transforms rank-reduced generalized eigenproblems into full-ranked standard eigenequations.

ABSTRACT

We consider to model matrix time series based on a tensor CP-decomposition. Instead of using an iterative algorithm which is the standard practice for estimating CP-decompositions, we propose a new and one-pass estimation procedure based on a generalized eigenanalysis constructed from the serial dependence structure of the underlying process. To overcome the intricacy of solving a rank-reduced generalized eigenequation, we propose a further refined approach which projects it into a lower-dimensional full-ranked eigenequation. This refined method improves significantly the finite-sample performance of the estimation. The asymptotic theory has been established under a general setting without the stationarity. It shows, for example, that all the component coefficient vectors in the CP-decomposition are estimated consistently with certain convergence rates. The proposed model and the estimation method are also illustrated with both simulated and real data; showing effective dimension-reduction in modelling and forecasting matrix time series.

Motivation & Objective

• To address overparametrization in high-dimensional matrix time series modeling by achieving effective dimension reduction.
• To overcome the drawbacks of iterative algorithms like ALS, including slow convergence and sensitivity to initial values.
• To develop a one-pass estimation procedure that avoids iterations and leverages serial dependence structure in the data.
• To establish asymptotic theory for CP-decomposition under general, non-stationary settings.
• To improve finite-sample performance by transforming rank-reduced generalized eigenproblems into full-ranked eigenequations via projection.

Proposed method

• The method constructs a generalized eigenanalysis problem from the serial dependence structure of the matrix time series, replacing iterative optimization.
• It introduces a refined approach that projects a rank-reduced generalized eigenequation into a lower-dimensional full-ranked eigenequation, simplifying computation.
• The estimation procedure is one-pass, avoiding the need for initialization or multiple iterations common in ALS.
• The method is grounded in tensor canonical polyadic (CP) decomposition of the $p \times q \times n$ tensor formed by frontal slices of the matrix time series.
• Asymptotic theory is developed without requiring stationarity, establishing consistency and convergence rates for component coefficient vectors.
• The approach is validated through simulation and real data, showing effective dimension reduction and improved forecasting performance.

Experimental results

Research questions

• RQ1 Can a one-pass estimation procedure be developed for tensor CP-decomposition of matrix time series that avoids the convergence and initialization issues of iterative methods?
• RQ2 How can serial dependence in matrix time series be exploited to construct a generalized eigenanalysis framework for CP-decomposition?
• RQ3 What is the finite-sample performance of a refined projection method that transforms rank-reduced generalized eigenproblems into full-ranked standard eigenequations?
• RQ4 Does the proposed method achieve consistent estimation of component vectors under non-stationary settings?
• RQ5 How does the proposed method compare to existing Tucker-based or iterative CP-decomposition approaches in terms of forecasting accuracy and dimension reduction?

Key findings

• The proposed one-pass estimation procedure achieves consistent estimation of all component coefficient vectors in the CP-decomposition with established convergence rates, even without stationarity.
• The refined projection method significantly improves finite-sample performance by transforming complex rank-reduced generalized eigenproblems into solvable full-ranked eigenequations.
• Simulation results show that the method outperforms both standard ALS and random initialization approaches, with lower mean squared error across various $p$, $q$, and $n$ configurations.
• In the real data example, the method effectively reduces dimensionality and improves forecasting accuracy compared to baseline models.
• The method demonstrates robustness across different matrix sizes and noise levels, with consistent performance gains in RMSE metrics.
• The asymptotic theory confirms the validity of the estimator under general, non-stationary settings, broadening applicability beyond traditional factor models.