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.

翻译：本文提出了一种用于矩阵值时间序列的模态可加因子模型，该模型通过可加结构捕捉行特定与列特定的潜在效应，相比Tucker和CP因子模型等乘法框架具有更强的灵活性。在MAFM中，每个观测值可分解为行因子分量、列因子分量与噪声项，从而允许沿不同模态的变异来源被独立建模。我们开发了一种计算高效的两阶段估计流程：首先采用模态内积特征分解进行初始化，随后通过补空间投影交替子空间估计算法进行迭代优化。该方法的核心创新在于，正交补投影在估计每个载荷空间时能完全消除跨模态干扰。我们在适当条件下建立了估计因子载荷矩阵的收敛速率，进一步推导了载荷矩阵估计量的渐近分布，并构建了一致的协方差估计量，从而形成了一套数据驱动的统计推断框架，可用于置信区间构建与假设检验。作为一项具有独立价值的技术贡献，我们建立了相依矩阵时间序列二次型的矩阵Bernstein不等式。在仿真数据与真实数据上的数值实验证明了所提方法相较于现有方法的优越性。