Modewise Additive Factor Model for Matrix Time Series

TL;DR

This paper proposes a flexible additive factor model for matrix time series that captures mode-specific effects, introduces an efficient estimation procedure, and provides statistical inference tools with theoretical guarantees.

Contribution

The paper introduces the Modewise Additive Factor Model (MAFM), a novel framework that models matrix time series with additive mode-specific effects, along with an efficient estimation and inference methodology.

Findings

The proposed MAFM outperforms existing models in simulations.

The estimation procedures achieve desirable convergence rates.

The method provides valid confidence intervals and hypothesis tests.

Abstract

We introduce a Modewise Additive Factor Model (MAFM) for matrix-valued time series that captures row-specific and column-specific latent effects through an additive structure, offering greater flexibility than multiplicative frameworks such as Tucker and CP factor models. In MAFM, each observation decomposes into a row-factor component, a column-factor component, and noise, allowing distinct sources of variation along different modes to be modeled separately. We develop a computationally efficient two-stage estimation procedure: Modewise Inner-product Eigendecomposition (MINE) for initialization, followed by Complement-Projected Alternating Subspace Estimation (COMPAS) for iterative refinement. The key methodological innovation is that orthogonal complement projections completely eliminate cross-modal interference when estimating each loading space. We establish convergence rates for the estimated factor loading matrices under proper conditions. We further derive asymptotic distributions for the loading matrix estimators and develop consistent covariance estimators, yielding a data-driven inference framework that enables confidence interval construction and hypothesis testing. As a technical contribution of independent interest, we establish matrix Bernstein inequalities for quadratic forms of dependent matrix time series. Numerical experiments on synthetic and real data demonstrate the advantages of the proposed method over existing approaches.