## 2 Conventional Diagonal Loading Methods
### The Improved GLC Method  
#### 3.2.1 Full Loading of the Covariance Matrix  
Since the GLC can improve the CMOS by minimizing the error between the loaded covariance matrix \(\mathbf{R}_{\text{GLC}}\)\(\in\)\(\mathbb{C}^{M\times M}\) and \(\mathbf{R}_{\text{EXP}}\), we chose it as the basic model. Firstly, the diagonal loading matrix was replaced by the full loading matrix, which can reduce the components of the target source and the error of the off-diagonal elements. The full loading-based covariance matrix is defined as follows:  
\[ \begin{split}\mathbf{R}_{\text{IGLC}}&=\mathbf{\alpha} \mathbf{R}_{\text{LM1}}+\mathbf{\beta}\mathbf{R}_{\text{LM2}}\\ &=[\alpha_{1},\alpha_{2},\alpha_{3}][\mathbf{R}_{\text{XX}}, \mathbf{R}_{\text{white}},\mathbf{I}]^{T}-[\beta_{1},\beta_{2},\cdots,\beta_{ P}][\mathbf{R}_{\text{SS},1},\mathbf{R}_{\text{SS},2},\cdots,\mathbf{R}_{\text{SS},P}]^{T} \\ &=\alpha_{1}\tilde{\mathbf{R}}_{\text{XX}}+\alpha_{2}\mathbf{R}_ {\text{white}}+\alpha_{3}\mathbf{I}-\sum\limits_{m=1}^{P}\beta_{m}\mathbf{R}_ {\text{SS},m}\end{split}  \tag{10}\]  
where \(\mathbf{\alpha}\) = [\(\alpha_{1}\),\(\alpha_{2}\),\(\alpha_{3}\)] and \(\beta\) = [\(\beta_{1}\),\(\beta_{2}\),...,\(\beta_{P}\)] are the vectors of the non-negative shrinkage parameters. \(\mathbf{R}_{\text{LM1}}\) = [\(\mathbf{R}_{\text{XX}}\), \(\mathbf{R}_{\text{white}}\), \(\mathbf{I}\)]\({}^{T}\), \(\mathbf{R}_{\text{LM2}}\) = [\(\mathbf{R}_{\text{SS},1}\),\(\mathbf{R}_{\text{SS},2}\),...,\(\mathbf{R}_{\text{SS},P}\)]\({}^{T}\). \(\mathbf{R}_{\text{white}}\)\(\in\)\(\mathbb{C}^{M\times M}\) is the basic covariance matrix of white noise and \(\mathbf{R}_{\text{SS},m}\)\(\in\)\(\mathbb{C}^{M\times M}\) is the covariance matrix corresponding to the \(m\)th eigenvalue of \(\mathbf{R}_{\text{XX}}\), which can be approximatively regarded as the covariance matrix of the \(m\)th source signal. The superscripted \"\(T\)\" indicates the transpose operation. \(P\) is the number of the significantly large eigenvalues [22; 23]. In Equation (10), \(\mathbf{R}_{\text{white}}\) is used to reduce the influence of the off-diagonal elements, \(\mathbf{I}\) is used to realize the diagonal loading and \(\mathbf{R}_{\text{SS},m}\) is used to reduce the components of the target source. Among them, \(\mathbf{R}_{\text{white}}\) and \(\mathbf{R}_{\text{SS},m}\) need to be solved. Here, we use the noise subspace related to the smallest eigenvalue to estimate \(\mathbf{R}_{\text{white}}\). Thus, \(\mathbf{R}_{\text{white}}\) and \(\mathbf{R}_{\text{SS},m}\) can be estimated as \(\mathbf{R}_{\text{white}}\)\(\approx\)\(\lambda_{M}\)\(\mathbf{v}_{M}\)\(\mathbf{v}_{M}^{H}\) and \(\mathbf{R}_{\text{SS},m}\)\(=\)\(\lambda_{m}\)\(\mathbf{v}_{m}\)\(\mathbf{v}_{m}^{H}\), respectively. Where \(\lambda_{1}\)\(\geq\)\(\lambda_{2}\)\(\geq\)\(\ldots,\)\(\geq\)\(\lambda_{m}\)\(\geq\)\(\ldots,\)\(\geq\)\(\lambda_{M}\) are the eigenvalues of \(\mathbf{R}_{\text{xx}}\), \(\mathbf{v}_{M}\)\(\in\)\(\mathbb{C}^{M\times 1}\) and \(\mathbf{v}_{m}\)\(\in\)\(\mathbb{C}^{M\times 1}\) are the  
Figure 1: The framework of the proposed method. eigenvectors related to \(\lambda_{M}\) and \(\lambda_{m}\), respectively. Hence, the optimization of the IGLC is built as follows:  
\[\underset{\alpha,\beta}{\text{min}}\text{E}\Big{\{}\left\|\mathbf{R}_{\text{ IGLC}}-\mathbf{R}_{\text{IN}}\right\|_{2}^{2}\Big{\}}=\underset{\alpha,\beta}{\text{min}} \text{E}\Bigg{\{}\left\|\alpha_{1}\mathbf{R}_{\text{XX}}+\alpha_{2}\mathbf{R}_ {\text{white}}+\alpha_{3}\mathbf{I}-\sum\limits_{m=1}^{P}\beta_{m}\mathbf{R}_ {\text{SS},m}-\mathbf{R}_{\text{IN}}\right\|_{2}^{2}\Bigg{\}} \tag{11}\]  
However, the covariance matrix \(\mathbf{R}_{\text{IN}}\) of the interference-plus-noise is unknown. Here, we use the uncertain set-based method [24; 25] to estimate \(\mathbf{R}_{\text{IN}}\). The uncertain set-based method uses spatial power (also called Capon power) to estimate the covariance matrix of each sound source.