Abstract

Existing methods of vector autoregressive model for multivariate time

series analysis make use of low-rank matrix approximation or Tucker decomposition to reduce the dimension of the over-parameterization issue. In this paper,

we propose a sparse Tucker decomposition method with graph regularization for

high-dimensional vector autoregressive time series. By stacking the time-series

transition matrices into a third-order tensor, the sparse Tucker decomposition is

employed to characterize important interactions within the transition third-order

tensor and reduce the number of parameters. Moreover, the graph regularization is employed to measure the local consistency of the response, predictor and

temporal factor matrices in the vector autoregressive model. The two proposed

regularization techniques can be shown to more accurate parameters estimation.

A non-asymptotic error bound of the estimator of the proposed method is established, which is lower than those of the existing matrix or tensor based methods

when the Tucker core is sufficiently sparse.

A proximal alternating linearized

minimization algorithm is designed to solve the resulting model and its global

convergence is established under very mild conditions. Numerical experiments

on synthetic data and real-world datasets are carried out to verify the superior

performance of the proposed method over existing state-of-the-art methods.

Key words and phrases: Graph regularization, high-dimensional vector autore- gression, proximal alternating linearized minimization algorithm, sparse Tucker decomposition

Information

Preprint No.SS-2026-0065
Manuscript IDSS-2026-0065
Complete AuthorsSijia Xia, Michael K. Ng, Xiongjun Zhang
Corresponding AuthorsXiongjun Zhang
Emailsxjzhang@ccnu.edu.cn

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Acknowledgments

The authors would like to thank the Associate Editor and the two anonymous referees for their constructive suggestions and comments, which helped

improve the quality of the paper. This research was partially supported by

GDSTC: Guangdong and Hong Kong Universities “1+1+1” Joint Research

Collaboration Scheme UICR0800008-24, National Key Research and Development Program of China under Grant 2024YFE0202900, RGC GRF

12300125, Joint NSFC and RGC N-HKU769/21, Hubei Provincial Natural Science Foundation of China under Grant 2025AFB966, Fundamental

Research Funds for the Central Universities under Grant XJ2026000601,

and Academy of Frontier Interdisciplinary Research, Central China Normal

University (JC2026PT-002).

Supplementary Materials

The supplementary materials contain detailed solutions to the subproblems in PALM, the proofs of Theorems 1 and 2, additional numerical ex-

periments for simulation and real data, and discussions about parameters

analysis, choice of lag order, different initialization, choices of Tucker rank,

constructions of graphs, ablation analysis, the extensions to uniform noise.

Supplementary materials are available for download.