## 2 Classifier Design
### Proposed PL-MSMM  
In this study, we introduced pinball loss into the multiclass SMM--namely, the PL-MSMM. The PL-MSMM is an anti-noise sensitive matrix form classifier, which effectively handles multiclass fault diagnosis tasks. Given a \(k\) class matrix training dataset \(\{\mathbf{X}_{i},y_{i}\}_{i=1}^{n}\in\{\mathcal{X},\mathcal{Y}\}\), the objective function of the PL-MSMM can be defined as follows:  
\[ \begin{split}&\min_{\mathcal{W}\xi}~{}\frac{1}{2}\parallel \mathcal{W}\parallel_{F}^{2}+\tau\sum_{j=1}^{k}\left\|\mathbf{W}_{j}\right\|_ {*}+\frac{C}{n}\sum_{i=1}^{n}\xi_{i},\\ &\text{s.t.}~{}\Delta(\hat{y}_{i},y_{i})+\langle\mathcal{W}, \delta\mathbf{\Psi}(\mathbf{X}_{i},\hat{y}_{i},y_{i})\rangle\leq\xi_{i}\\ &\text{and}~{}-p(\Delta(\hat{y}_{i},y_{i})+\langle\mathcal{W}, \delta\mathbf{\Psi}(\mathbf{X}_{i},\hat{y}_{i},y_{i})\rangle)\leq\xi_{i}\\ &\text{for}~{}\forall(\hat{y}_{1},\hat{y}_{2},...,\hat{y}_{n})\in \mathcal{Y}^{n},\end{split}  \tag{4}\]  
where \(\mathcal{W}\in\mathbb{R}^{p\times q\times k}\) denotes the tensor form of the regression parameter, \(\parallel\mathcal{W}\parallel_{F}\) denotes the Frobenius norm of \(\mathcal{W}\), and \(\left\{\mathbf{W}_{j}\right\}_{j=1}^{k}\) denotes the matrix-form hyperplane of the \(j\)th class data. \(C\) represents the loss term parameter, and \(\tau\) is positive values used to constrain the kernel norm. \(p\) is the pinball loss parameter, \(\xi\) represents a sequence of slack variables for the hinge loss, and \(\delta\mathbf{\Psi}(\mathbf{X}_{i},\hat{y}_{i},y_{i})\) represents the discrepancy in feature mappings between an arbitrary label \(\hat{y}_{i}\) and the truth label \(y_{i}\) for the input \(\mathbf{X}_{i}\). \[\delta\mathbf{\Psi}(\mathbf{X}_{i},\hat{y}_{i},y_{i})=\mathbf{\Psi}(\mathbf{X }_{i},\hat{y}_{i})-\mathbf{\Psi}(\mathbf{X}_{i},y_{i}), \tag{5}\]where the feature map \(\mathbf{\Psi}(\mathbf{X},j)\in\mathbb{R}^{p\times q\times k}\) denotes a sparse tensor with zero elements except for \(\mathbf{\Psi}_{:,:,j}=\mathbf{X}\). Equation (4) contains \(n\) slack variables. Owing to the multiclass pinball loss and nuclear norm, the optimization problem is non-smooth and non-differentiable. To address this, we can merge \(n\) slack variables into a single variable and construct a framework based on the alternating direction method of multipliers (ADMM). The ADMM methodology, renowned for its adeptness in handling non-smooth and non-differentiable optimization problems, emerges as a suitable alternative for effectively navigating the intricacies inherent to the problem at hand [27]. By exploiting the independence of each estimated label, the \(n\) slack variable \(\xi_{i}\) in (4) can be reduced to a single slack variable \(\xi\), \(\xi=\frac{1}{n}\sum_{i=1}^{n}\xi_{i}\), which denotes the equivalent upper bound of the inequality under all constraints. This allows us to rewrite the objective function as follows:  
\[ \begin{split}&\min_{\mathcal{W}\xi}\ \frac{1}{2}\parallel \mathcal{W}\parallel_{F}^{2}+C\xi+\tau\sum_{c=1}^{k}\|\mathbf{\mathrm{W}}_{c}\|_{*} \\ &\text{s.t.}\ \frac{1}{n}\sum_{i=1}^{n}\left\{\Delta(\hat{g}_{i},y_{ i})+\langle\mathcal{W},\delta\mathbf{\Psi}(\mathbf{X}_{i},\hat{g}_{i},y_{i})\rangle \right\}\leq\xi\\ &\text{and}\ \ -p\frac{1}{n}\sum_{i=1}^{n}\left\{(\Delta(\hat{g}_{ i},y_{i})+\langle\mathcal{W},\delta\mathbf{\Psi}(\mathbf{X}_{i},\hat{g}_{i},y_{i}) \rangle)\right\}\leq\xi\\ &\text{for}\ \forall(\hat{y}_{1},\hat{y}_{2},...,\hat{y}_{n})\in \mathcal{Y}^{n}.\end{split}  \tag{6}\]  
The number of slack variables in (6) can be reduced to one by increasing the number of constraints on \(|\mathcal{Y}|^{n}\), all the constraints sharing a single slack variable \(\xi\). By rewriting the objective function in an unconstrained form, the above equation can be expressed as follows:  
\[