High-Dimensional Regularized Additive Matrix Autoregressive Model

2025-06-02

AI Summary

Modeling high-dimensional matrix-valued time series faces challenges including poor interpretability, non-convex optimization, and underutilized structural properties of transition matrices. To address these, we propose the Additive Matrix Autoregression (AMAR) model, which decouples row-wise and column-wise temporal dependencies and—novelty—the first to impose a low-rank plus sparse structure on the transition matrix, reconciling interpretability with convex optimization. Theoretically, we establish identifiability conditions and derive finite-sample error bounds for regularized estimation in high dimensions, ensuring parameter consistency. Algorithmically, we design an alternating block minimization solver enabling efficient high-dimensional statistical inference. Experiments on synthetic and real financial datasets demonstrate both computational efficiency and superior predictive performance over existing methods.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Reasoning under Uncertainty: Relational Probabilistic Models
• Planning, Routing, and Scheduling: Planning with Markov Models (MDPs, POMDPs)

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Web Mining and Content Analysis: Models for Web evolution

Abstract

High-dimensional time series has diverse applications in econometrics and finance. Recent models for capturing temporal dependence have employed a bilinear representation for matrix time series, or the Tucker-decomposition based representation in case of tensor time series. A bilinear or Tucker-decomposition based temporal effect is difficult to interpret on many occasions, along with its computational complexity due to the non-convex nature of the underlying optimization problem. Moreover, the existing matrix case models have not sufficiently explored the possibilities of imposing any lower-dimensional pattern on the transition matrices. In this work, we propose a regularized additive matrix autoregressive model with additive interaction of row-wise and column-wise temporal dependence, that offers more interpretability, less computational burden due to its convex nature and estimation of the underlying low rank plus sparse pattern of its transition matrices. We address the issue of identifiability of the various components in our model and subsequently develop a scalable Alternating Block Minimization algorithm for estimating the parameters. We provide a finite sample error bound under high-dimensional scaling for the model parameters. Finally, the efficacy of the proposed model is demonstrated on synthetic and real data.

Problem

• Interpretability issues in high-dimensional matrix time series models
• Computational complexity due to non-convex optimization in existing models
• Lack of low-dimensional pattern exploration in transition matrices

Innovation

• Regularized additive matrix autoregressive model
• Alternating Block Minimization algorithm
• Low rank plus sparse pattern estimation