## 3 MTR Algorithm Based on Multi-Layer Sparse Structure  
### Multi-Layer Mtr  
The purpose of MTR is to learn a mapping function \(h:\mathbb{R}^{d}\rightarrow\mathbb{R}^{Q}\) to map the input space \(\mathbf{X}\in\mathbb{R}^{d}\) to the output space \(\mathbf{Y}\in\mathbb{R}^{Q}\), \(d\) and \(Q\) are the dimensionality of input and output spaces, respectively. Without loss of generality, take the basic linear MTR model as an example, which can be expressed as \(\mathbf{y}=\mathbf{W}\mathbf{x}+\mathbf{b}\), where \(\mathbf{y}=\big{[}y_{1},y_{2},\cdots,y_{Q}\big{]}^{\top}\in\mathbb{R}^{Q}\) is the multi-variate targets, \(\mathbf{x}\in\mathbb{R}^{d}\) is the input, and \(\mathbf{W}=\big{[}\mathbf{w}_{1},\mathbf{w}_{2},\cdots,\mathbf{w}_{Q}\big{]} ^{\top}\in\mathbb{R}^{Q\times d}\) is the coefficient matrix corresponding to the model, which reflects the linear correlation between the input features and the output target. \(\mathbf{w}_{i}\in\mathbb{R}^{d}\) is the parameter vector corresponding to the target output \(\mathbf{y}_{i}\), and \(\mathbf{b}\in\mathbb{R}^{Q}\) is the bias. Given the training set \(\{(\mathbf{x}_{i},\mathbf{y}_{i})\}_{i=1}^{N}\), the following penalty objective function can be established:  
\[\mathbf{W}^{*}=\underset{\mathbf{W}}{\text{argmin}}\frac{1}{N}\|\mathbf{Y}- \mathbf{W}\mathbf{X}-\mathbf{B}\|_{F}^{2}+\lambda\|\mathbf{W}\|_{F}^{2}. \tag{1}\]  
where \(\|\mathbf{Y}-\mathbf{W}\mathbf{X}-\mathbf{B}\|_{F}^{2}\) is the loss function and \(\mathbf{X}=[\mathbf{x}_{1},\mathbf{x}_{2},\cdots,\mathbf{x}_{N}]\), \(\mathbf{Y}=[\mathbf{y}_{1},\mathbf{y}_{2},\cdots,\mathbf{y}_{N}]\) and \(\mathbf{B}=[\cdots,\mathbf{b},\cdots]\in\mathbb{R}^{Q\times N}\), \(N\) is the number of training samples and \(\lambda\) is the penalty parameter corresponding to coefficient matrix \(\mathbf{W}\). \(\|\mathbf{W}\|_{F}^{2}\) is the Frobenius norm of matrix of \(\mathbf{W}\), which can be computed by \(\mathbf{W}\), and \(tr\Big{(}\mathbf{A}^{\top}\mathbf{A}\Big{)}\) is the regularization parameter. Equation (1) is a straightforward form of single-output linear ridge regression extended to multiple outputs. Although it can realize the optimization of multiple outputs simultaneously, it does not consider the correlation between different subentry cost variables in WSM data, and thus cannot explore the correlation knowledge in the original information of different subentry costs to improve the overall fitting performance of the model. To effectively explore the intrinsic correlation of different subentry costs on WSM data, the latent variable space is introduced to form the corresponding multi-layer structure based on Equation (1). Compared with the traditional method of applying various constraints to the coefficient matrix \(\mathbf{W}\) to explore the correlation between target outputs, the multi-layer multi-output structure based on latent variable space can effectively separate the redundant relationship between feature inputs and structural correlation between outputs, which makes the learned coefficient matrix more interpretable and flexible. On the other hand, the multi-layer multi-output structure is more conducive to decoupling and expressing the complex input-output relationships in high-dimensional data. By sparse coding of inter-target correlations in the latent variable space, the correlations among WSM subentry costs can be better explored and expressed. The MTR model is constructed as follows:  
\[\underset{\mathbf{W},\mathbf{S}}{\text{min}}\frac{1}{N}\|\mathbf{Y}-\mathbf{S }\mathbf{Z}\|_{F}^{2}+\lambda_{1}\|\mathbf{W}\|_{F}^{2}+\lambda_{2}\|\mathbf{ S}^{\top}\|_{2,1}. \tag{2}\]where \(\mathbf{Z}=[\mathbf{z}_{1},\mathbf{z}_{2},\cdots,\mathbf{z}_{N}]=\mathbf{W}\mathbf{X} +\mathbf{B}\in\mathbb{R}^{Q\times N}\), and \(\mathbf{z}_{i}=\mathbf{W}\mathbf{x}_{i}+\mathbf{b}\in\mathbb{R}^{Q}\) contains the latent variables in the latent space, \(\mathbf{S}\in\mathbb{R}^{Q\times Q}\) is the structure matrix designed to explore correlations between different targets.