Matrix Autoregressive Model

[Definition] (Chen, Xiao & Yang’21) Given matrix-variate time series $\boldsymbol{X}_{t}\in\mathbb{R}^{M\times N},t=1,\ldots,T$ , then the $d$ th-order matrix autoregressive (MAR( $d$ )) model takes the form of

$\boldsymbol{X}_{t}=\sum_{k=1}^{d}\boldsymbol{A}_{k}\boldsymbol{X}_{t-k}\boldsymbol{B}_{k}^\top+\boldsymbol{E}_{t}$

where $\boldsymbol{A}_{k}\in\mathbb{R}^{M\times M},\boldsymbol{B}_{k}\in\mathbb{R}^{N\times N},k=1,\ldots,d$ are the coefficient matrices.

[Questions] Regarding the following optimization problem:

$\min_{\boldsymbol{A},\boldsymbol{B}}~\frac{1}{2}\sum_{t=d+1}^{T}\left\|\boldsymbol{X}_{t}-\sum_{k=1}^{d}\boldsymbol{A}_{k}\boldsymbol{X}_{t-k}\boldsymbol{B}_{k}^\top\right\|_{F}^{2}$

How to obtain the global minimum with respect to the coefficient matrices? Is alternating least squares (ALS) the best solution algorithm?

[Note] According to Nash equilibrium, the alternating minimization procedure is guaranteed to globally converge to a block coordinate-wise minimum (Xu & Yin’13).

Matrix Autoregressive Model

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