## 2 Preliminary
### Tensor-Train Decomposition  
Assuming that there is a high-order tensor \(\mathcal{X}\), a sequence of third-order core tensors \(\mathcal{G}^{(1)},\mathcal{G}^{(2)},\dots,\mathcal{G}^{(N)}\) are used to express \(\mathcal{X}\), which is called TT decomposition. \(\mathcal{X}\in\mathbb{R}^{h_{1}\times 2\times\dots\times h_{n}}\) is given by  
\[\mathcal{X}=\ll\mathcal{G}^{(1)},\mathcal{G}^{(2)},\dots,\mathcal{G}^{(N)}\gg \tag{13}\]  
where \(\mathcal{X}\in\mathbb{R}^{h_{1}\times h_{2}\times\dots\times h_{n}}\), \(\mathcal{G}\in\mathbb{R}^{h_{n-1}\times h_{n}\times h_{n}}\), \(R_{0}=R_{N}=1\). The notation \(\ll\gg\) represents the approximation of the core tensor sequence to the tensor \(\mathcal{X}\). \(l_{0},l_{1},\dots,l_{N}\) are used to define the dimension of \(\mathcal{X}\). \(R_{0},R_{1},\dots,R_{N}\) determine the size of each core tensor, which is called TT rank. Smaller TT rank yields a smaller overall dimension of the series \(\mathcal{G}\), which will greatly reduce the complexity. In addition, the \((i_{1},i_{2},\dots,i_{N})\)-th element of the tensor \(\mathcal{X}\) can be expressed as  
\[x_{i_{1},i_{2},\dots,i_{N}}=\prod_{n=1}^{N}G_{i_{n}}^{(n)}. \tag{14}\]  
Figure 1: NOMS decoding network. where \(G_{i_{1}}^{(1)},G_{i_{2}}^{(2)},\ldots,G_{i_{N}}^{(N)}\) is the sequence of slices obtained from each core tensor. There is \(\mathcal{O}(dIR^{2})\) spatial complexity of TT format, where \(R\) is the maximum value of the TT ranks, \(I\) is the maximum value of modes in \(I_{1},I_{2},\ldots,I_{n}\), and \(d\) is the order of the tensor. For completeness, the background on TT decomposition is explained in the following. For further studies, readers can refer to [8; 23]. TT decomposition employs a sequence of singular value decomposition (SVD) or the cross approximation algorithm. Assume that we want to approximate a given tensor \(\mathcal{A}\) by a tensor \(\mathcal{X}\), that is \(\mathcal{X}\approx\mathcal{A}\), with a prescribed accuracy \(\epsilon\). We then need to find the optimal rank and the optimal series of core tensors, \(\mathcal{G}^{(1)},\mathcal{G}^{(2)},\ldots,\mathcal{G}^{(N)}\) under the condition \(\parallel\mathcal{X}-\mathcal{A}\parallel_{F}\leq\epsilon\). It follows that \(N\) consecutive shaping and SVD calculations constitute the computation of the TT decomposition. The detailed steps of TT decomposition via recursive SVD are as follows. First, we validate the TT rank, \(R_{0},R_{1},\ldots,R_{N}\), with below rules: (1) the length of rank should match with the dimension of \(\mathcal{A}\), (2) consecutive factors should have matching ranks, (3) boundary conditions dictate that the head and tail rank set to 1. Second, we set \(\mathcal{X}_{1}=\mathcal{X}\), and we will find the optimal \(\mathcal{G}^{(1)},\mathcal{G}^{(2)},\ldots,\mathcal{G}^{(N-1)}\) one by one. We flatten the unfolding tensor \(\mathcal{X}_{1}\) of the remaining factors, where \(\mathcal{X}_{1}\in\mathbb{R}^{I_{1}\times I},I=I_{2}\times\ldots\times I_{n}\). We get 1-st TT factor \(U\) in \(\mathcal{X}_{1}=USV^{T}\) with \(U\) as a \(I_{1}\times I_{1}\) matrix and \(V\) as an \(I\times I_{1}\) matrix. The head TT core \(\mathcal{G}^{(1)}=U\) is given by the \(U\) indexed by \(R_{0},I_{1},R_{1}\). SVD is chosen for its numerical stability in the TT algorithm for binary analysis in this step. Then, the new unfolding tensor for the remaining factors is \(\mathcal{X}_{1}=SV^{T}\). Repeat the second step \(N-2\) times, and reshape \(\mathcal{X}_{1}\) such that its rows are indexed by \(R_{n-1},I_{n},\forall n\in[N-1]\) and its columns by remaining portion. Then we will get each core, \(\mathcal{G}^{(1)},\mathcal{G}^{(2)},\ldots,\mathcal{G}^{(N-1)}\) which are reshaped into \(\mathcal{G}^{(n)}\in\mathbb{R}^{R_{n-1}\times I_{n}\times R_{n}}\) tensors, \(\forall n\in[N-1]\). Finally, the last factor \(\mathcal{G}^{(N)}\in\mathbb{R}^{R_{N-1}\times I_{N}\times R_{N}}\) is obtained by \(\mathcal{X}_{1}\).