# Multi-Domain Communication Systems and Networks: A Tensor-Based Approach
## 3 Simultaneous Signal Processing Across Domains
### Tensor-Based Receiver Designs
#### 3.2.2 Application of Tensor Train Decomposition and Tensor Networks  
Algorithms which allow to perform various tensor operations without reshaping the tensors provide an important feature as matrix or vector transformation of the tensors is not desirable in many applications. Very often, large data are stored using Tensor Train (TT) decomposition for reducing storage complexity [28; 32]. In TT format, a higher order tensor is written in terms of a set of sparsely connected lower order tensors called cores. For a tensor \(\mathcal{A}\) of size \(J_{1}\times J_{2}\times\cdots\times J_{M}\), the TT decomposition is specified by:  
\[\mathcal{A}_{J_{1},\ldots,J_{M}}=\sum_{r_{0},r_{1},\ldots,r_{N}}\mathcal{A}_{r _{0},i_{1},r_{1}}^{(1)}\cdot\mathcal{A}_{r_{1},j_{2},r_{2}}^{(2)}\cdots \mathcal{A}_{r_{M-1},j_{M},r_{M}}^{(M)} \tag{19}\]  
where each core \(\mathcal{A}^{(m)}\) is a third order tensor of size \(R_{m-1}\times J_{m}\times R_{m}\) and \(R_{m}\) denote the TT ranks with \(R_{0}=R_{M}=1\) and \(R_{m}\geq 1\) for \(m=1,\ldots,M-1\)[28]. Since the TT ranks determine the storage consumption of the tensor, the structure of the core tensors \(\mathcal{A}^{(m)}\) with low TT ranks is exploited for reducing the storage complexity. In such a process, rather than storing the entire tensor, only the tensor cores along with the information on the modes to be contracted are stored. Hence, any mathematical operation to be performed on the tensor should be able to act on the cores itself without reconstructing the entire tensor. This restricts any reshaping of the data. A tensor in TT format can be graphically represented through a Tensor Network (TN) containing nodes and edges [32]. Each core is represented by a node, and a contraction between two cores is represented by connecting the nodes through an edge. The free edges emerging from a core correspond to the modes which are not contracted. Any mathematical computation between two tensors can be expressed in terms of their TNs. For instance, contracted product between two different tensors can be achieved using TNs by connecting the free edges corresponding to the modes to be contracted of the two tensors. A detailed description and examples of different tensor operations using TN representation is provided in [32]. Consider the system model presented for order 3 input output tensors and order 6 channel tensors in Figure 1. A TN representation of this model is illustrated in Figure 5. It shows the representation of the Einstein product between the third order input \(\mathcal{X}\in\mathbb{C}^{I_{1}\times I_{2}\times I_{3}}\) and the sixth order channel \(\mathcal{H}\in\mathbb{C}^{I_{1}\times I_{2}\times I_{3}\times I_{1}\times I_{2 }\times I_{3}}\) in TT format. The operation \(\mathcal{Y}=\mathcal{H}*_{3}\mathcal{X}\) can be written in terms of the cores of \(\mathcal{H}\) and \(\mathcal{X}\) as  
\[\mathcal{Y}_{i_{1},i_{2},i_{3}}=\sum_{i_{1},i_{2},i_{3}}\Big{(}\sum_{r_{0},r_ {1},\ldots,r_{0}}\underbrace{\mathcal{H}_{r_{0},i_{1},r_{1}}^{(1)}\cdot \mathcal{H}_{r_{1},j_{2},r_{2}}^{(2)}\cdots\mathcal{H}_{r_{N},j_{N}}^{(6)}}_{ \text{orange nodes}}\Big{)}\cdot\Big{(}\sum_{r_{0},r_{1},r_{2},r_{3},\ldots,r_{ 0}}\underbrace{\mathcal{X}_{r_{0},i_{1},i_{1}}^{(1)}\cdot\mathcal{X}_{r_{1},j _{2},r_{3}}^{(2)}\cdot\mathcal{X}_{r_{2},i_{3},i_{3}}^{(3)}}_{\text{green nodes}}\Big{)} \tag{20}\]  
The summation over indices \(i_{1},i_{2},i_{3}\) in (20) is reflected in the edges connecting the green and the orange nodes in Figure 5. Specific algorithms to compute the [PERSON] product, which act directly on the cores of the tensor train format are presented in [33]. More recently, the concept of tensor networks has been also considered for developing error-correcting codes such as a generalization of polar codes. The notion of polar codes was first introduced in [34] as a class of capacity-achieving codes for binary memoryless symmetric channels. The use of such channel codes can significantly enhance the transmission reliability of wireless communication systems.