# Structural-Missing Tensor Completion for Robust DOA Estimation with Sensor Failure +
## 3 Structural-Missing Tensor Completion
### Spatio-Temporal Dimension Augmentation for Missing Element Dispersion  
Due to the existence of structurally missing elements in the incomplete tensor signal \(\mathbf{\mathcal{\hat{X}}}\), the conventional low-rank regularization on an incomplete tensor with randomly missing elements becomes infeasible. Thus, to guarantee effective low-rank regularization on the tensor signal, we first design a spatio-temporal dimension augmentation method to disperse the structurally missing elements into the higher-dimensional space.  
In particular, considering that different dimensions of the tensor signal present strong spatio-temporal relevance, properly augmenting these dimensions will enhance the capability of parameter retrieval, and also helps to disperse the structurally missing elements. The tensorial Hankelization approach is known for regularly embedding spatial/temporal delay information, which can be adopted for spatio-temporal dimension augmentation of the structural-missing tensor signal \(\mathbf{\mathcal{\hat{X}}}\). To be more specific, as shown in Figure 2, \(\mathbf{\mathcal{\hat{X}}}\) can be transformed into a six-dimensional (6D) dimensional-augmented Hankel tensor \(\mathbf{\mathcal{H}}\in\mathbb{C}^{\intercal_{1}\times(M-\intercal_{1}+1)\times \intercal_{2}\times(N-\intercal_{2}+1)\times\intercal_{3}\times(T-\intercal_{ 3}+1)}\), defined as  
\[\mathbf{\mathcal{H}}=fold_{(\intercal_{1})}(\mathbf{\mathcal{\hat{X}}}\times_{1}\mathbf{S} _{1}\times_{2}\mathbf{S}_{2}\times_{3}\mathbf{S}_{3}), \tag{7}\]  
where the folding operator \(fold_{(\intercal_{1})}\) organizes a \(\intercal_{1}(L_{1}-\intercal_{1}+1)\times\intercal_{2}(L_{2}-\intercal_{2}+ 2)\times\cdots\times\intercal_{t}(L_{I}-\intercal_{I}+1)\) tensor into another \(\intercal_{1}\times(L_{1}-\intercal_{1}+1)\times\intercal_{2}\times(L_{2}- \intercal_{2}+1)\times\cdots\times\intercal_{t}\times(L_{I}-\intercal_{I}+1)\) tensor, \(\intercal_{i}\) denotes the duplication factor that expands the \(i\)-th dimension from the size of \(L_{i}\) to \(\intercal_{i}(L_{i}-\intercal_{i}+1)\), \(\times_{i}\) denotes the mode-\(i\) tensor-matrix product, and \(\mathbf{S}_{i}\in\{0,1\}^{\intercal_{i}(L_{i}-\intercal_{i}+1)\times L_{i}}\) denotes the duplication matrices for \(i=1,2,\cdots,I\) with \(I\) being the tensor order (For the structural-missing tensor signal \(\mathbf{\mathcal{\hat{X}}}\), we have \(I=3\), but such an operation can be easily generalized to 3D cubic array with higher-dimensional tensor signals). Here, the \((p,q)\)-th element of \(\mathbf{S}_{i}\) can be denoted by  
\[S_{i_{(p,q)}}=\begin{cases}1&q=\lfloor p/\tau_{i}\rfloor+mod(p,\tau_{i}),\\ 0&\text{others},\end{cases} \tag{8}\]  
where \(\lfloor\cdot\rfloor\) denotes the rounding down operator, and \(mod(\cdot,\cdot)\) denotes the residual operator. Since the size of the expanded dimensions should be a positive integer, we have \(\tau_{i}<L_{i}\).  
By performing spatio-temporal dimension augmentation of the structural-missing tensor signal as demonstrated in (7), its missing fibers or slices are now dispersed across six dimensions of the dimensional-augmented Hankel tensor \(\mathbf{\mathcal{H}}\), such that the low-rank tensor completion can be subsequently incorporated for its completion.