Given a third-order tensor $\mathcal{X}\in\mathbb{R}^{I_1\times I_2\times I_3}$, we want to find an approximation tensor $\hat{\mathcal{X}}$ of $\mathcal{X}$ with $R$ rank-one components, and some entries of $\hat{\mathcal{X}}$ are given. Thus the optimization problem can be formulated as: \begin{equation*} \begin{array}{cl} \arg\min\limits_{\hat{\mathcal{X}}}\quad& \|\mathcal{X}-\hat{\mathcal{X}}\|\\ \mathrm{s.t.}\quad&\hat{\mathcal{X}} = \sum\limits_{r=1}^{R}\mathbf{a}_{r}^{(1)}\circ\mathbf{a}_{r}^{(2)}\circ\mathbf{a}_{r}^{(3)},\\ \quad&\hat{\mathcal{X}}_{ijk}=\mathcal{X}_{ijk},\quad (i,j,k)\in E. \end{array} \end{equation*} Where $E$ is an index subset, and $\mathbf{a}_{r}^{(l)}\in\mathbb{R}^{I_l}$ ($l$=1,2,3). How to solve this optimization problem? Is there any software for using?

As we know, a similar optimization problem called CP decomposition of a tensor, \begin{equation*} \begin{array}{cl} \arg\min\limits_{\hat{\mathcal{X}}}\quad& \|\mathcal{X}-\hat{\mathcal{X}}\|\\ \mathrm{s.t.}\quad&\hat{\mathcal{X}} = \sum\limits_{r=1}^{R}\mathbf{a}_{r}^{(1)}\circ\mathbf{a}_{r}^{(2)}\circ\mathbf{a}_{r}^{(3)} \end{array} \end{equation*} can be solve using the alternating least squares (ALS) method, and serval software can perform this efficiently.


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