# Structural-Missing Tensor Completion for Robust DOA Estimation with Sensor Failure +
## 3 Structural-Missing Tensor Completion
### Dimensional-Augmented Hankel Tensor Completion  
Thanks to the imposed spatio-temporal dimension augmentation, the resultant incomplete Hankel tensor \(\mathbf{\mathcal{H}}\) has sufficiently dispersed missing elements along with augmented spatio-temporal dimensions. As such, the augmented Hankel tensor \(\mathbf{\mathcal{H}}\) can be completed based on the low CP rank regularization (Note that, the Tucker-based tensor completion causes a problem of model mismatch in this scenario. Specifically, the Tucker decomposition yields a core tensor with uncertain Tucker ranks, which is inconsistent with the tensor signal model. Meanwhile, Tucker ranks should be manually initialized for Tucker decomposition, which causes performance deterioration if they are incorrectly set. Moreover, CPD based on the residual error minimization reconstructs a completed tensor, where the observed elements are modified. However, the observed elements in the tensor signal are expected to remain unchanged to avoid model deviations. As such, these tensor completion methods are not applicable to robust DOA estimation), which is characterized by the following optimization problem  
\[\begin{split}\min_{\mathbf{\mathcal{H}}}&\tau_{\text{CP} }(\mathbf{\mathcal{H}})\\ \text{s.t.}&\text{P}_{\Omega}(\mathbf{\mathcal{H}})= \text{P}_{\Omega}(\mathbf{\mathcal{H}}),\end{split} \tag{9}\]  
where the optimization variable \(\mathbf{\mathcal{H}}\in\mathbb{C}^{\tau_{1}\times(M-\tau_{1}+1)\times\tau_{2} \times(N-\tau_{2}+1)\times\tau_{3}\times(T-\tau_{3}+1)}\) is the completed dimensional-augmented Hankel tensor, \(\tau_{\text{CP}}\) denotes the CP rank of a tensor, \(\Omega\) is the index set of missing elements in \(\mathbf{\mathcal{H}}\), and \(\Omega\) is the complement set of \(\Omega\), i.e., the index set of the observed elements in \(\mathbf{\mathcal{H}}\). The proposed dimensional-augmented Hankel tensor completion problem in (9) minimizes the CP rank of \(\mathbf{\mathcal{H}}\) under the constraint that all observed elements in \(\mathbf{\mathcal{H}}\) are kept in \(\mathbf{\mathcal{H}}\). It is worth mentioning that the above optimization problem (9) is non-convex due to the NP-hardness of determining tensor ranks.  
To solve the non-convex dimensional-augmented Hankel tensor completion problem (9), it is converted into a convex tensor nuclear norm minimization problem. The reason for this lies in that the nuclear norm is proved to be the tightest convex relaxation of  
Figure 2: Illustration of the proposed structural-missing tensor completion.  
the rank function. Specifically, the dimensional-augmented Hankel tensor nuclear norm minimization problem can be represented as  
\[\begin{split}\underset{\boldsymbol{\hat{\mathcal{H}}}}{\text{min}} \ \|\boldsymbol{\hat{\mathcal{H}}}\|_{*}=\sum_{j=1}^{6}\alpha_{j}\|[ \boldsymbol{\hat{\mathcal{H}}}]_{j}\|_{*}\\ \text{s.t.}\ \ \ \text{P}_{\bar{\Omega}}\big{(}\boldsymbol{ \hat{\mathcal{H}}}\big{)}=\text{P}_{\bar{\Omega}}(\boldsymbol{\mathcal{H}}), \end{split} \tag{10}\]  
where \(\|\cdot\|_{*}\) denotes the nuclear norm, \([\,\cdot\,]_{j}\) denotes mode-\(j\) tensor unfolding, and \(\alpha_{j}\geqslant 0\) is the coefficient satisfying \(\sum_{j=1}^{6}\alpha_{j}=1\). Note that, the matrix nuclear norm is proven to be the tightest relaxation of the rank function, and the tensor can be characterized by its unfolding matrices along respective dimensions. As such, the tensor nuclear norm serves as a convex combination of the nuclear norms of its matrix unfoldings, which makes it possible to solve the above problem with convex optimization approaches.