# 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
In practice, no assumption of the diagonality exists for the covariance of all deep speaker embeddings, and all kinds of covariances of all deep speaker embeddings are far from diagonal. It means that only i-vector satisfies the prerequisite of the GLASSO-PLDA. The characteristics in the two covariance structures of the i-vector and deep speaker embeddings lead to different results in applying the regularization method, the GLASSO, to the PLDA. In practice, as described in Section 5.3, only the i-vector, which satisfies the prerequisite, showed the performance improvements when using the GLASSO-PLDA. All the deep speaker embeddings, which have \(\mathbf{\Phi}_{w}\) and \(\mathbf{\Phi}_{w}^{-1}\) that are far from diagonal, showed the performance degradations when using the GLASSO-PLDA. Our goal of the use of the GLASSO is to reduce the estimation errors in the precision matrix. We aim to remove small noises in the precision matrix raised by the estimation errors. In the case of close-to-diagonal precision, like i-vector, the GLASSO-PLDA exhibited the performance improvement. Therefore, the GLASSO can remove many of the small noises effectively when the precision matrix is close to diagonal. However, in the case of far-from-diagonal precision, like deep speaker embeddings, the GLASSO-PLDA demonstrated the performance degradation or failure of the estimation of the PLDA parameters. The GLASSO also shrinks the principal elements that have relatively larger values toward zero, on the contrary a too-large bias toward zero, when the precision matrix is far from diagonal. Thus, the loss from a high bias is greater than the gain from the low estimation error for that case. This result may imply that the GLASSO can improve the performance only if the covariance/precision matrix is close to diagonal. We assume that deep speaker embeddings also show the performance improvements with the GLASSO-PLDA, like i-vector, if we make the within-class covariance matrix \(\mathbf{\Phi}_{w}\) and the accompanying precision matrix \(\mathbf{\Phi}_{w}^{-1}\) close to diagonal. To make the \(\mathbf{\Phi}_{w}\) and \(\mathbf{\Phi}_{w}^{-1}\) of the deep speaker embeddings close to diagonal, we orthogonalize the deep speaker embeddings using principal component analysis  
Figure 1: The empirical within-class covariance matrix of (**a**) i-vectors, (**b**) d-vectors, (**c**) r-vectors, and (**d**) x-vectors. Figure 2: The empirical within-class precision matrix of (**a**) i-vectors, (**b**) d-vectors, (**c**) r-vectors, and (**d**) x-vectors. (PCA). The PCA transform makes the total covariance close to diagonal through orthogonalization. The diagonalization of the total covariance has the effect of making both within-class covariance and the accompanying precision matrix close to diagonal. We use the orthogonalized deep speaker embeddings for the GLASSO-PLDA instead of the original embeddings (denote as PCA-GLASSO-PLDA). We do not reduce the dimensionality in the PCA transform to avoid information loss. It is important to check whether \(\mathbf{\Phi}_{w}\) and \(\mathbf{\Phi}_{w}^{-1}\) actually become closer to diagonal after the PCA transform. Figures 3 and 4 reveal the within-class covariance and within-class precision matrix estimated from the transformed embeddings (denoted as \(\mathbf{\Phi}_{w\_pa}\) and \(\mathbf{\Phi}_{w\_pa}^{-1}\)), respectively. After applying the PCA transform, all the within-class covariance/precision matrix (corresponding to \(\mathbf{\Phi}_{w\_pca}\) and \(\mathbf{\Phi}_{w\_pa}^{-1}\), respectively) become closer to diagonal. The diagonality \(\delta\) of \(\mathbf{\Phi}_{w\_pa}\) and \(\mathbf{\Phi}_{w\_pa}^{-1}\) is 0.1509 (Figure 3a) and 0.1935 (Figure 4a) for the d-vectors, 0.2262 (Figure 3b) and 0.364 (Figure 4b) for the r-vectors, and 0.1728 (Figure 3c) and 0.1827 (Figure 4c) for the x-vectors, respectively. As described in Section 5.3, the orthogonalized deep speaker embeddings exhibited performance improvements with the GLASSO-PLDA (corresponding to the PCA-GLASSO-PLDA), as in the i-vector.