# Weak Fault Detection of Tapered Rolling Bearing Based on Penalty Regularization Approach
## 2 Augmented Huber Non-Convex Penalty Regularization
### Convexity Condition  
In this section, the convexity of the proposed OCF will be proved to consider how to set the parameter \(\tau\) and the relationship between matrix \(\mathbf{A}\) and matrix \(\mathbf{B}\). If \(y\in\mathbb{R}^{M},\mathbf{A}\in\mathbb{R}^{M\times N}\) is a matrix, the regularization term parameter \(\lambda>0\) and the proposed OCF \(F(x)=\frac{1}{2}\left\|y-\mathbf{A}x\right\|_{2}^{2}+\lambda\phi_{B}(x)\) is convex, then the matrix \(\mathbf{B}^{T}\mathbf{B}\) satisfies  
\[\mathbf{B}^{T}\mathbf{B}\leq\frac{1}{\lambda}\mathbf{A}^{T}\mathbf{A} \tag{19}\]  
**Proof 2. ** According to \(s_{B}(x)=\underset{v\in\mathbb{R}}{\min}\Big{\{}\left\|v\right\|_{1}+\frac{1}{ 2}\left\|\mathbf{B}(x-v)\right\|_{2}^{2}\Big{\}}\) and \(\phi_{B}(x)=\left\|x\right\|_{1}-s_{B}(x)\), the \(F(x)\) can be rewritten as  
\[F(x)=\frac{1}{2}\left\|y-\mathbf{A}x\right\|_{2}^{2}+\lambda\left\|x \right\|_{1}-s_{B}(x)\] (20) \[=\frac{1}{2}\left\|y-\mathbf{A}x\right\|_{2}^{2}+\lambda\left  
For convexity condition \(B^{T}B\leq\frac{1}{\lambda}A^{T}A\), it can be rewritten as \(B^{T}B=(\tau/\lambda)A^{T}A\), in which the parameter \(\tau\) satisfies \(\tau\leq 1\). Hence, given a matrix \(A\), the matrix \(B\) could be set as \(B=\sqrt{\tau/\lambda}A,0\leq\tau\leq 1\). The parameter \(\tau\) controls the non-convexity of the AHF \(\phi_{B}(x)\). If \(\tau=0\), then \(B=0\) and function \(\phi_{B}(x)\) reduces to the L1-norm. If \(\tau=1\), then \(B^{T}B=(\tau/\lambda)A^{T}A\) and the function \(\phi_{B}(x)\) are maximally non-convex. Here, the nominal range of parameter \(\tau\) is discussed by a simulated discrete signal. Given a simulated systematic signal of  
\[ \begin{array}{l}f(t)=f_{1}(t)+\text{Noise}\\ =M_{0}\sin(2\pi f_{1}t)+M_{1}\cos(2\pi f_{2}t)+\text{Noise}\end{array}  \tag{21}\]  
where the amplitudes of \(M_{0}\) and \(M_{1}\) are 1 and 2, respectively, the signal length is \(N=100\) with the frequencies \(f_{1}=0.25\) and \(f_{2}=0.4\). For the de-noising experiment, the Gaussian noise with \(\sigma=1.5\) is added to acquire the simulated systematic signal. The systematic signal \(f\)(t) is approximately sparse in the frequency domain, as illustrated in Figure 3, which can be represented by \(f=Ax\), where \(A\) is an over-sampled inverse discrete Fourier transform and \(x\) is a sparse vector of Fourier coefficients. Specifically, matrix \(A\) is defined as \(A_{m,n}=(1/\sqrt{N})\exp(j(2\pi/N)mn)\), \(m=0,1,\cdots,M-1\); \(n=0,1,\cdots,N-1\), with \(M=100\) and \(N=256\). Assuming that matrix \(A^{T}\) is the complex conjugate transpose of matrix \(A\) and it can be calculated that \(B^{T}B\) is a diagonal matrix. In order to obtain a nominal range of parameter \(0\leq\tau\leq 1\), we vary the parameter \(\tau\) from 0.05 to 0.95 (by increments of 0.05) and calculate each root-mean-square error (RMSE) between the actual value \(f_{1}(t)\) and its estimated value \(f_{1}(t)\), and also we calculate the optimized coefficients (or Fourier coefficients amplitude) at frequencies \(f_{1}=0.25\) and \(f_{2}=0.4\) for each parameter \(\tau\). The RMSE values and optimized coefficients for each parameter \(\tau\) are shown in Figure 4. From Figure 4a, it can be seen that the larger value of parameter \(\tau\) would get a lower value of RMSE, and the optimal parameters \(\tau\) are concentrated in the \(0.8\leq\tau\leq 0.9\) range. As a matter of fact, the smaller value of parameters \(\tau\) would make the estimated signal noisier and less sparse. From Figure 4b, it is found that the optimized coefficients at \(f_{1}=0.25\) increase and then slightly decrease with increasing parameter \(\tau\); meanwhile, the optimized coefficients at \(f_{1}=0.4\) increase as parameter \(\tau\) increases. From the point of view that the significant coefficients are not underestimated and get a clearer signal \(f_{1}(t)\), the optimal parameters \(\tau\) are also concentrated in the \(0.8\leq\tau\leq 0.9\) range. Therefore, in practice the nominal range of parameter \(\tau\) is set to \(0.8\leq\tau\leq 0.9\).