## 2 The Tensor Modeling of Structured Sparse SAR Imaging
### Multiway Sparsity  
One-dimensional expansion of a tensor is defined as stacking all elements along the mode-1, e.g., \(\mathbf{y}=\mathit{vec}\left(\mathcal{Y}\right)=\mathit{vec}\left(\mathbf{Y}_{ \left(1\right)}\right)\), where _mode_ is considered as the order of a tensor. Then the Tucker decomposition of \(\mathcal{Y}\) is expressed as \(\mathcal{S}\times_{1}\mathbf{A}_{1}\times_{2}\mathbf{A}_{2}\times_{3}\cdots\times_{N }\mathbf{A}_{N}\), where \(\times_{n}\) denotes the mode-\(n\) product of tensor with matrix [37], which can be written as  
\[\mathbf{y}=\left(\mathbf{A}_{N}\otimes\mathbf{A}_{N-1}\otimes\cdots\otimes\mathbf{A}_{1} \right)\mathbf{s}. \tag{10}\]  
It is indicated from (10) that a Tucker decomposition of a multiway data is equivalent to a linear representation with Kronecker dictionary.  
**Definition 1** (Multiway sparsity).: _If \(\mathbf{y}\) in the Tucker model (10) is \(K\)-sparse with respect to \(\mathbf{A}=\left(\mathbf{A}_{N}\otimes\mathbf{A}_{N-1}\otimes\cdots\otimes\mathbf{A}_{1}\right)\), then \(\mathcal{Y}\) is \(K\)-sparse with respect to the factor matrices \(\mathbf{A}_{n}\left(n=1,2,\ldots,N\right)\). The columns of \(\mathbf{A}_{n}\) are the atoms along each mode._  
For an \(N\)-dimensional signal, if it has a \(K\)-sparse representation, the core tensor of Tucker decomposition has \(K\) nonzero elements, i.e.,  
\[\mathcal{Y}=\sum_{k=1}^{K}s_{\hat{x}_{1}^{k},\hat{x}_{2}^{k},\ldots,\hat{x}_{ N}^{k}}\mathbf{A}_{1}\left(\cdot;\hat{x}_{1}^{k}\right)\circ\mathbf{A}_{2}\left(\cdot; \hat{x}_{2}^{k}\right)\circ\cdots\circ\mathbf{A}_{N}\left(\cdot;\hat{x}_{N}^{k} \right). \tag{11}\]  
where \"\(\circ\)\" is the outer product operator. Because the nonzero entries of scatter coefficients are non-uniformly distributed and always gather together in the practical application of SAR imaging, this feature results in the highly structured echo signal, by which more rapid algorithm may be constructed.  
**Definition 2** (Multiway structured sparsity).: _A multiway signal \(\mathcal{Y}\) is called \(\left(K_{1},K_{2},\ldots,K_{N}\right)\)-structured sparse, if the factor matrices of its Tucker decomposition only need \(K_{n}\left(n=1,\ldots,N\right)\) columns to compute, i.e.,_  
\[\mathcal{Y}=\mathcal{S}\times_{1}\mathbf{A}_{1}\times_{2}\mathbf{A}_{2} \times_{3}\cdots\times_{N}\mathbf{A}_{N}, \tag{12}\] \[s.t.\ \ \ s_{i_{1},i_{2},\ldots,i_{N}}=0\,\forall\left(i_{1},i_{2}, \ldots,i_{N}\right)\
otin\left\{\mathcal{I}_{1},\mathcal{I}_{2},\ldots,\mathcal{ I}_{N}\right\},\]  
_where \(\mathcal{I}_{n}=\left[i_{n}^{1},i_{n}^{2},\ldots,i_{n}^{K_{n}}\right]\) indicates a subset of the indices for mode-\(n\)\(\left(n=1,2,\ldots,N\right)\)._  
By Definition 2, the nonzero elements of core tensor are concentrated in a subtensor \(\mathcal{S}\left(\mathcal{I}_{1},\mathcal{I}_{2},\ldots,\mathcal{I}_{N}\right)\). Then the relationship between multiway structured sparsity and the vector signal sparsity can be deduced: \(\mathbf{y}=\mathit{vec}\left(\mathcal{Y}\right)\) is \(K\)-sparse (\(K=K_{1}\times K_{2}\times\cdots\times K_{N}\)) with respect to Kronecker dictionary \(\mathbf{A}=\left(\mathbf{A}_{N}\otimes\mathbf{A}_{N-1}\otimes\cdots\otimes\mathbf{A}_{1}\right)\). It should be noted that the definition of _structured sparsity_ here is different from the definition of _block sparsity_ used for one-dimensional signal. For the latter, a coefficient vector is called block \(K\)-sparse if it consists of a series of concatenate segments, named blocks, and the number of segments without nonzero Euclidean norm is less than \(K\)[38; 39].