# Weak Fault Detection of Tapered Rolling Bearing Based on Penalty Regularization Approach
## 2 Augmented Huber Non-Convex Penalty Regularization
### The Augmented Huber Non-Convex Penalty Regularization  
In this section, before the elicitation of the proposed AHNPR algorithm, the following definitions and proposition for convex analysis should be reviewed first. The Huber equation is defined through two piecewise functions [31; 32], i.e.,  
\[s(x)=\left\{ \begin{array}{ll}\frac{1}{2}x^{2},&|x|\leq 1\\ \|x\|-\frac{1}{2},&|x|\geq 1\end{array} \right. \tag{6}\]  
The plot of Huber function \(s(x)\) is illustrated in Figure 1.  
Let \(b\) denote a scale and the scaled Huber function (SHF) can be defined as  
\[s_{b}(x)=\frac{1}{b^{2}}s(b^{2}x),b\
eq 0 \tag{7}\]  
For \(b=0\), i.e., \(s_{0}(x)=0\). For \(b\
eq 0\), the scaled Huber function is given by  
\[s_{b}(x)=\left\{ \begin{array}{ll}\frac{1}{2}b^{2}x^{2},&|x|\leq 1/b^{2}\\ |x|-\frac{1}{2b^{2}},&|x|\geq 1/b^{2}\end{array} \right. \tag{8}\]  
Note that \(0\leq s_{b}(x)\leq|x|,\forall x\in R\), i.e., \(\lim\limits_{b\rightarrow\infty}s_{b}(x)=|x|\) and \(\lim\limits_{b\to 0}s_{b}(x)=0\). For the definition of SHF, the SHF could be rewritten through a minimization form, i.e.,  
Figure 1: The Huber function \(s(x)\). \[s_{b}(x)=\min\limits_{v\in\mathbb{R}}\biggl{\{}\|v\|_{1}+\frac{1}{2}b^{2}(x-v)^{2} \biggr{\}} \tag{9}\]  
**Proof 1. ** According to the SHF, i.e., \(s_{b}(x)=\frac{1}{b^{2}}s(b^{2}x),b\
eq 0\), Equation (9) can be rewritten as  
\[ \begin{array}{l}s_{b}(x)=\min\limits_{v\in\mathbb{R}}\Bigl{\{}\|v\|_{1}+ \frac{1}{2}{(b^{2}x-v)}^{2}\Bigr{\}}/b^{2}\\ =\min\limits_{v\in\mathbb{R}}\biggl{\{}\|b^{2}v\|_{1}+\frac{1}{2}{(b^{2}x-b^{2 }v)}^{2}\Bigr{\}}/b^{2}\\ =\min\limits_{v\in\mathbb{R}}\bigl{\{}\|v\|_{1}+\frac{1}{2}{b^{2}(x-v)}^{2} \Bigr{\}}\end{array}  \tag{10}\]  
Let \(\mathbf{B}\) denote a matrix and the generalized Huber function (GHF) can be defined as  
\[s_{B}(x)=\min\limits_{v\in\mathbb{R}}\biggl{\{}\|v\|_{1}+\frac{1}{2}\|\mathbf{B}(x -v)\|_{2}^{2}\biggr{\}} \tag{11}\]  
In summary, if matrix \(\mathbf{B}\) evolves into a scale, i.e., \(\mathbf{B}=b\), then the GHF reduces to the SHF, \(s_{B}(x)=s_{b}(x)\) for all \(x\). If \(\mathbf{B}\) is a matrix and \(\mathbf{B}^{T}\mathbf{B}=\mathbf{I}\) is a diagonal, then the GHF is separable, and \(s_{B}(x)\) is the sum of the SHF \(s_{b}(x)\), i.e., \(\mathbf{B}^{T}\mathbf{B}=\text{Diag}(b_{1},b_{2},\cdots,b_{N})\Rightarrow s_{B}(x)= \sum\limits_{N}s_{b_{n}}(x_{n})\). In contrast, if \(\mathbf{B}^{T}\mathbf{B}\
eq\mathbf{I}\) is a non-diagonal matrix, the GHF is non-separable. Based on the definitions above, the new non-separable non-convex penalty function can be derived using the GHFs. The new non-separable non-convex penalty (NNP) function \(\phi(x)\) is defined as  
\[\phi(x)=\left\{ \begin{array}{ll}|x|-\frac{1}{2}x^{2},&|x|\leq 1\\ \frac{1}{2},&|x|\geq 1\end{array} \right. \tag{12}\]  
Therefore, according to the definition of Huber function, the function \(\phi(x)\) can be expressed as  
\[\phi(x)=|x|-s(x) \tag{13}\]  
where \(s(x)\) is the Huber function. Similarly, the plot of NNP function \(\phi(x)\) is illustrated in Figure 2.  
Let \(b\) denote a scale of function and the scaled NNP function can be defined as  
\[\phi_{b}(x)=|x|-s_{b}(x) \tag{14}\]  
Figure 2: The non-separable non-convex penalty function. where \(s_{b}(x)\) is SHF. For \(b=0\), i.e., \(\phi_{0}(x)=|x|\). For \(b\
eq 0\), the scaled NNP function is given by  
\[\phi_{b}(x)=\left\{ \begin{array}{ll}|x|-\frac{1}{2}b^{2}x^{2},&|x|\leq 1/b^{2} \\ \frac{1}{2b^{2}},&|x|\geq 1/b^{2}\end{array} \right. \tag{15}\]  
Let \(\mathbf{B}\) denote a matrix and the NNP function \(\phi(x)\) will evolve into an augmented Huber function (AHF) \(\phi_{B}(x)\), i.e.,  
\[\phi_{B}(x)=\left\|x\right\|_{1}-s_{B}(x) \tag{16}\]  
where \(s_{B}(x)\) is the generalized Huber function (GHF).