# Regularized Within-Class Precision Matrix Based PLDA in Text-Dependent Speaker Verification
## 3 Graphical Least Absolute Shrinking and Selection Operator (GLASSO)
### Prerequisite: Close-to-Diagonal Within-Class Covariance/Precision Matrix  
Even though a common normality assumption exists regarding the embeddings for both PLDA and GLASSO, we do not consider the normality assumption in our research. As mentioned in Section 2.1 and, we used four kinds of speaker embeddings, that is i-vector, d-vector, r-vector, and x-vector. Among these embeddings, only the i-vector satisfies the normality assumption. The i-vector has demonstrated performance improvement with both PLDA and GLASSO-PLDA. However, all deep speaker embeddings used in our research, which have no assumption of normality, also exhibited performance improvement with the PLDA. In contrast, the deep speaker embeddings show performance degradation or failure of the estimation with the GLASSO-PLDA. The experimental results will be described in Section 5.3. Some research has investigated making the deep speaker embeddings follow a normal distribution [54; 55; 56]. However, we found that deep speaker embeddings obtained using these methods are still not suitable for the GLASSO-PLDA (see Appendix A), and the embeddings obtained by [55] were corrupted and lost discriminative power. From this result, we conclude that the absence of the normality assumption on the embeddings does not matter, and regard all embeddings used in our experiments as following a multivariate normal distribution. We focus on the diagonality of the within-class covariance/precision matrix. Close-to-diagonal covariance matrix of normal variables relates to that the covariates are not severely dependent, which is the condition for the GLASSO, as mentioned in Section 3.2. In addition, the closer the covariance matrix is to the diagonal, the closer the accompanying precision matrix is somewhat to the diagonal. For a quantitative comparison, we define the degree of the diagonality (denote as \(\delta\)) of the matrix \(\boldsymbol{\Theta}\) as the ratio of the \(L_{1}\) norm of the covariance \(\|\boldsymbol{\Theta}\|_{1}\) to the \(L_{1}\) norm of the diagonal elements \(\|\text{diag}(\boldsymbol{\Theta})\|_{1}\):  
\[\delta=\frac{\|\text{diag}(\boldsymbol{\Theta})\|_{1}}{\|\boldsymbol{\Theta} \|_{1}}. \tag{13}\]  
The higher \(\delta\) means that \(\boldsymbol{\Theta}\) is more close to the diagonal matrix, and \(\delta\) satisfies \(0\leq\delta\leq 1\). As mentioned above, the prerequisite of the GLASSO-PLDA is that the empirical within-class covariance matrix \(\boldsymbol{\Phi}_{w}\) and accompanying precision matrix \(\boldsymbol{\Phi}_{w}^{-1}\) should be close to diagonal. Therefore, it is important to check whether \(\boldsymbol{\Phi}_{w}\) and \(\boldsymbol{\Phi}_{w}^{-1}\) of each kind of embedding are close to diagonal. Figures 1 and 2 illustrate \(\boldsymbol{\Phi}_{w}\) and \(\boldsymbol{\Phi}_{w}^{-1}\) of four kinds of embeddings, respectively. The diagonality \(\delta\) of \(\boldsymbol{\Phi}_{w}\) and \(\boldsymbol{\Phi}_{w}^{-1}\)is 0.2179 (Figure 1a) and 0.2 (Figure 2a) for the i-vectors, 0.0262 (Figure 1b) and 0.0696 (Figure 2b) for the d-vectors, 0.0522 (Figure 1c) and 0.0794 (Figure 2c) for the r-vectors, and 0.019 (Figure 1d) and 0.0916 for the x-vectors (Figure 2d), respectively. As you can see, only \(\mathbf{\Phi}_{w}\) and \(\mathbf{\Phi}_{w}^{-1}\) of the i-vector (Figures 1a and 2a) are close to diagonal, and the others (Figures 1b-d and 2b-d) are far from diagonal. The covariances of the i-vector are assumed to be zero, that is, the features consisting of the i-vector are independent. In most cases, the empirical covariance matrix is close to the diagonal matrix due to the estimation error. Therefore, both \(\mathbf{\Phi}_{w}\) and \(\mathbf{\Phi}_{w}^{-1}\) of the i-vectors are close to diagonal. In contrast, both \(\mathbf{\Phi}_{w}\) and \(\mathbf{\Phi}_{w}^{-1}\) of all deep speaker embeddings are far from diagonal, even considering the estimation error.