Split-and-Conquer: Distributed Factor Modeling for High-Dimensional Matrix-Variate Time Series

TL;DR

This work develops Split-and-Conquer, a distributed framework for high-dimensional matrix-valued time series via a matrix factor model Y_t = R F_t C^T + E_t. It partitions data column-wise or row-wise across nodes, estimates local loadings with two-dimensional tensor PCA (α-PCA), and aggregates them with a final PCA to obtain global loading matrices while preserving the latent matrix structure. The authors establish consistency and asymptotic normality of the loading estimators under mild conditions, extend the framework to unit-root nonstationary series with improved convergence rates, and demonstrate substantial computational gains with competitive accuracy in simulations and real data (Fama–French stock returns and OECD CPI). The approach offers practical scalability for big-data matrix-valued time series, with robust data-allocation strategies and consistent covariance estimation. Overall, Split-and-Conquer delivers a scalable, theoretically-grounded method for dimension reduction and forecasting in diverse, heterogeneous matrix-variate time series settings.

Abstract

In this paper, we propose a distributed framework for reducing the dimensionality of high-dimensional, large-scale, heterogeneous matrix-variate time series data using a factor model. The data are first partitioned column-wise (or row-wise) and allocated to node servers, where each node estimates the row (or column) loading matrix via two-dimensional tensor PCA. These local estimates are then transmitted to a central server and aggregated, followed by a final PCA step to obtain the global row (or column) loading matrix estimator. Given the estimated loading matrices, the corresponding factor matrices are subsequently computed. Unlike existing distributed approaches, our framework preserves the latent matrix structure, thereby improving computational efficiency and enhancing information utilization. We also discuss row- and column-wise clustering procedures for settings in which the group memberships are unknown. Furthermore, we extend the analysis to unit-root nonstationary matrix-variate time series. Asymptotic properties of the proposed method are derived for the diverging dimension of the data in each computing unit and the sample size $T$. Simulation results assess the computational efficiency and estimation accuracy of the proposed framework, and real data applications further validate its predictive performance.

Figures (5)

• Figure 1: Boxplots of the average and maximum column space distances $\mathcal{D}_1(\widehat{\mathbf{R}}_i)$, $\mathcal{D}_2(\widehat{\mathbf{R}}_i)$, $\mathcal{D}_1(\widehat{\mathbf{C}}_j)$, and $\mathcal{D}_2(\widehat{\mathbf{C}}_j)$, defined in Equation (\ref{['D1D2']}). Data are generated from model (\ref{['model1']}), ${\mathbf Y}_t={\mathbf R}{\mathbf F}_t{\mathbf C}^\top+{\mathbf E}_t$, where ${\mathbf F}_t$ and ${\mathbf E}_t$ follow the AR(1) processes in Equation (\ref{['generatedata1']}) with $(k,r)=(3,3)$ and $\psi=0.1$, and entries of ${\mathbf R}$ and ${\mathbf C}$ are drawn independently from $\mathcal{U}(-1, 1)$. We consider four dimension settings, $(p,q)\in\{(50,50),(50,100),(100,50),(100,100)\}$, and four sample size settings, $T\in\{0.5pq,1.0pq,1.5pq,2.0pq\}$. Results are based on 200 repetitions, with $\alpha=0$, $s_1=s_2=5$, and even partitions.
• Figure 2: Density histograms of the first element of $\widehat{\Sigma}_{{\mathbf R}_1}^{-1/2}\sqrt{\overline{m}T}(\widehat{{\mathbf R}}^1-{\mathbf R}^1{\mathbf H}_{\mathbf R})^\top$. Red curves denote the standard normal distribution. Data are generated from model (\ref{['model1']}), ${\mathbf Y}_t={\mathbf R}{\mathbf F}_t{\mathbf C}^\top+{\mathbf E}_t$, where ${\mathbf F}_t$ and ${\mathbf E}_t$ follow the matrix normal distributions in Equation (\ref{['generatedata2']}) with $(k,r)=(3,3)$ and $(p,q)=(200,200)$, and entries of ${\mathbf R}$ and ${\mathbf C}$ are drawn independently from $\mathcal{U}(-1, 1)$. We consider four sample size settings, $T\in\{0.5pq,1.0pq,1.5pq,2.0pq\}$. Results are based on 1000 repetitions, with $\alpha=0$, $s_1=s_2=3$, and even partitions.
• Figure 3: Average runtime (log s) for estimating loading matrices using our method and $\alpha$-PCA. Data are generated from model (\ref{['model1']}), ${\mathbf Y}_t={\mathbf R}{\mathbf F}_t{\mathbf C}^\top+{\mathbf E}_t$, where ${\mathbf F}_t$ and ${\mathbf E}_t$ follow the AR(1) processes in Equation (\ref{['generatedata1']}) with $(k,r)=(3,3)$ and $\psi=0.1$, and entries of ${\mathbf R}$ and ${\mathbf C}$ are drawn independently from $\mathcal{U}(-1, 1)$. Two settings are considered: In setting I, we fix $p=q=100$ and increase $T$ from 100 to 10,000; In setting II, we fix $T=100$ and increase $p=q$ from 50 to 1,000. Results are based on 200 repetitions, with $\alpha=0$, $s_1=s_2=5$, and even partitions.
• Figure B1: Boxplots of the average and maximum column space distances $\mathcal{D}_1(\widehat{\mathbf{R}}_i)$, $\mathcal{D}_2(\widehat{\mathbf{R}}_i)$, $\mathcal{D}_1(\widehat{\mathbf{C}}_j)$, and $\mathcal{D}_2(\widehat{\mathbf{C}}_j)$, defined in Equation (5.3). Data are generated from model (2.1), ${\mathbf Y}_t={\mathbf R}{\mathbf F}_t{\mathbf C}^\top+{\mathbf E}_t$, where ${\mathbf F}_t$ and ${\mathbf E}_t$ follow the AR(1) processes in Equation (5.2) with $(k,r)=(3,3)$ and $\psi=0.5$, and entries of ${\mathbf R}$ and ${\mathbf C}$ are drawn independently from $\mathcal{U}(-1, 1)$. We consider four dimension settings, $(p,q)\in\{(50,50),(50,100),(100,50),(100,100)\}$, and four sample size settings, $T\in\{0.5pq,1.0pq,1.5pq,2.0pq\}$. Results are based on 200 repetitions, with $\alpha=0$, $s_1=s_2=5$, and even partitions.
• Figure B2: Density histograms of the first element of $\widehat{\Sigma}_{{\mathbf R}_1}^{-1/2}\sqrt{\overline{m}T}(\widehat{{\mathbf R}}^1-{\mathbf R}^1{\mathbf H}_{\mathbf R})^\top$. Red curves denote the standard normal distribution. Data are generated from model (2.1), ${\mathbf Y}_t={\mathbf R}{\mathbf F}_t{\mathbf C}^\top+{\mathbf E}_t$, where ${\mathbf F}_t$ and ${\mathbf E}_t$ follow the matrix normal distributions in Equation (5.4) with $(k,r)=(3,3)$ and $(p,q)=(200,200)$, and entries of ${\mathbf R}$ and ${\mathbf C}$ are drawn independently from $\mathcal{U}(-1, 1)$. We consider four sample size settings, $T\in\{0.5pq,1.0pq,1.5pq,2.0pq\}$. Results are based on 1000 repetitions, with $\alpha=0$, $s_1=s_2=3$, and even partitions.

Theorems & Definitions (54)

• Proposition 1
• proof
• Theorem 1
• Theorem 2
• Theorem 3
• Remark 1
• Theorem 4
• Remark 2
• Theorem 5
• Theorem 6