# Modewise Additive Factor Model for Matrix Time Series

arXiv: https://arxiv.org/abs/2512.25025 · HTML: https://mlsanity.com/paper/2512.25025/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Dec 2025.

## Built on by (most cited first, 2 of 2)

- [Dual-Channel Tensor Neural Networks: Finite-Sample Theory and Conformal Structure Selection](https://mlsanity.com/paper/2605.19122/) (2026)
- [Factor Augmented High-Dimensional SGD](https://mlsanity.com/paper/2605.19291/) (2026)

## Abstract (arXiv)

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.