## 3 Estimation of Regularization Parameter
### Generalized Cross-Validation Method  
The randomized trace estimation was introduced into the GCV function to solve the difficulty of a trace calculation in the GCV function [13]. Then, this function for Equation (5) has the form  
\[GCV(\mu)\approx\frac{\left\|\mathbf{r}-\mathbf{h}\Big{(}\mathbf{h}^{T}\mathbf{ h}+\mu^{2}\mathbf{W_{m}}^{T}\mathbf{W_{m}}\Big{)}^{-1}\mathbf{h}^{T} \mathbf{r}\right\|_{2}^{2}}{\left\{n-\mathbf{R}^{T}\mathbf{h}\Big{(}\mathbf{h} ^{T}\mathbf{h}+\mu^{2}\mathbf{W_{m}}^{T}\mathbf{W_{m}}\Big{)}^{-1}\mathbf{h}^ {T}\mathbf{R}\right\}^{2}}, \tag{11}\]  
where \(\mathbf{R}\) is a random vector consisting of \(-1\) and \(1\), each with a probability of \(0.5\). The parameter \(\mu\) which minimizes (11) is the optimal regularization parameter \(\mu_{opt}\). This parameter \(\mu_{opt}\) can be determined by line search within a range for regularization parameter, and the process can be solved by using the CG method [13]. However, when using the CG method, an optimal range for regularization parameter is difficult to determine, and sometimes the optimal regularization parameter \(\mu_{opt}\) is not within the selected range. Moreover, the GCV method using the CG method is time-consuming. The GSVD of the matrix pair \([\mathbf{h},\mathbf{W_{m}}]\) can be used to compute the GCV function [5]. Based on the GSVD, the spectrum of \(\mathbf{h}\) can be obtained. The optimal linear search range can be determined by analyzing the spectrum of \(\mathbf{h}\)[9]. However, the calculation of GSVD is time-consuming and has a large memory requirement. When the scale of data is large, the calculation of GSVD is even impractical. Instead, the RGSVD algorithm is adopted. The RGSVD algorithm uses the randomized algorithm to provide a low-rank approximation of the GSVD [10]. Compared with the GSVD, the RGSVD can reduce computational costs and memory demands, and results with good accuracy can be obtained. Algorithm 2 shows the RGSVD algorithm [10], which is used in this study. The parameter \(q\) determines the accuracy and efficiency of the RGSVD algorithm. As the value of the parameter \(q\) increases, the result becomes more accurate and the computation time is longer. ```
1. Generate a \(q\times n\) Gaussian random matrix \(\mathbf{A}\). 2. Calculate the \(q\times m\) matrix \(\mathbf{Y}=\mathbf{A}\mathbf{h}\). 3. Calculate the \(m\times q\) orthonormal matrix \(\mathbf{Q}\) via QR factorization \(\mathbf{Y}^{T}=\mathbf{Q}\mathbf{R}\). 4. Form the \(n\times q\) matrix \(\mathbf{B}_{1}=\mathbf{h}\mathbf{Q}\) and the \(4m\times q\) matrix \(\mathbf{B}_{2}=\mathbf{W_{m}}\mathbf{Q}\). 5. Use \([\mathbf{U},\mathbf{V},\mathbf{W},\mathbf{C},\mathbf{S}]=\mathrm{gsvd}( \mathbf{B}_{1},\mathbf{B}_{2},0)\) to calculate the economy-sized GSVD of the matrix \(\mathrm{pair}\left[\mathbf{B}_{1},\mathbf{B}_{2}\right]\): \(\left[ \begin{array}{c}\mathbf{B}_{1}\\ \mathbf{B}_{2}\end{array} \right]\): \(\left[ \begin{array}{c}\mathbf{B}_{1}\\ \mathbf{B}_{2}\end{array} \right]\) = \(\left[ \begin{array}{c}\mathbf{U}\\ \mathbf{V}\end{array} \right]\) \(\left[ \begin{array}{c}\mathbf{C}\\ \mathbf{S}\end{array} \right]\) \(\mathbf{W}^{T}\). 6. Form the \(q\times m\) matrix \(\mathbf{X}=\mathbf{W}^{T}\mathbf{Q}^{T}\). 7. Note : \(\mathbf{h}\approx\mathbf{UCX}\) and \(\mathbf{W_{m}}\approx\mathbf{VSX}\). ```  
**Algorithm 2** Randomized generalized singular value decomposition (RGSVD) algorithm. In the RGSVD algorithm, \(\mathbf{C}=diag\big{(}c_{1},c_{2}\ldots,c_{q}\big{)}\in R^{q\times q}\) with  
\[0<c_{1}\leq c_{2}\leq\cdots\leq c_{q}<1, \tag{12}\]  
and \(\mathbf{S}=diag(s_{1},s_{2},\ldots,s_{2})\in R^{q\times q}\) with  
\[1>s_{1}\geq s_{2}\geq\cdots\geq s_{q}>0, \tag{13}\]  
\(c_{i}^{2}+s_{i}^{2}=1\), \(\forall i=1:q\), that is \(\mathbf{C}^{T}\mathbf{C}+\mathbf{S}^{T}\mathbf{S}=\mathbf{I}_{q}\).