# Regularized Within-Class Precision Matrix Based PLDA in Text-Dependent Speaker Verification
## 6 Discussion
### Evaluation in Text-Independent Speaker Verification  
We explain the difference between TD-SV and TI-SV in terms of variability. The total variability of samples \(\mathbf{\Sigma}\) can be decomposed as the sum of the between-class variability \(\mathbf{\Sigma}_{b}\) and the within-class variability \(\mathbf{\Sigma}_{w}\); that is, \(\mathbf{\Sigma}=\mathbf{\Sigma}_{b}+\mathbf{\Sigma}_{w}\). In TD-SV, \(\mathbf{\Sigma}_{b}\) contains the speaker variability \(\mathbf{\Sigma}_{spk}\) and phrase variability \(\mathbf{\Sigma}_{plbr}\), and \(\mathbf{\Sigma}_{w}\) contains other variabilities, such as channel variability \(\mathbf{\Sigma}_{ch}\) and residual variability \(\mathbf{\Sigma}_{c}\). Therefore, \(\mathbf{\Sigma}_{b}=\mathbf{\Sigma}_{spk}+\mathbf{\Sigma}_{plbr}\) and \(\mathbf{\Sigma}_{w}=\mathbf{\Sigma}_{ch}+\mathbf{\Sigma}_{c}\). In TI-SV, in contrast,\(\mathbf{\Sigma}_{b}\) contains only \(\mathbf{\Sigma}_{spk}\), and \(\mathbf{\Sigma}_{w}\) contains other variabilities \(\mathbf{\Sigma}_{plbr}\), \(\mathbf{\Sigma}_{ch}\), and \(\mathbf{\Sigma}_{c}\). Therefore, \(\mathbf{\Sigma}_{b}=\mathbf{\Sigma}_{spk}\) and \(\mathbf{\Sigma}_{w}=\mathbf{\Sigma}_{plbr}+\mathbf{\Sigma}_{ch}+\mathbf{\Sigma}_{c}\). To summarize, the difference between TD-SV and TI-SV is whether \(\mathbf{\Sigma}_{w}\) contains \(\mathbf{\Sigma}_{plbr}\) or not. For TD-SV, \(\mathbf{\Sigma}_{w}=\mathbf{\Sigma}_{ch}+\mathbf{\Sigma}_{c}\) does not contain phrase variability \(\mathbf{\Sigma}_{plbr}\). For TI-SV, in contrast, \(\mathbf{\Sigma}_{w}=\mathbf{\Sigma}_{plbr}+\mathbf{\Sigma}_{ch}+\mathbf{\Sigma}_{c}\) contains \(\mathbf{\Sigma}_{plbr}\). Therefore, the GLASSO regularization of the empirical within-class precision matrix \(\mathbf{\Phi}_{w}^{-1}\) does not affect \(\mathbf{\Sigma}_{plbr}\) for TD-SV but affects \(\mathbf{\Sigma}_{plbr}\) for TI-SV. For the evaluation of the task of TI-SV, we used the VoxCeleb1 dataset [(68)], which is divided into development (148,642 utterances from 1211 speakers) and test (4874 utterances from 40 speakers) sets. The average duration of utterances is 8.2 s. We split the development set into training (144,990 utterances from 1183 speakers) and validation (3652 utterances from 28 speakers; speaker id10357-id10384) sets. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{3}{*}{**Method**} & \multirow{3}{*}{**Embedding**} & \multicolumn{4}{c}{**EER (\%)**} \\ \cline{3-6}  & & & \multicolumn{2}{c}{**Evaluation**} \\ \cline{3-6}  & & & **Validation** & **Male** & **Female** \\ \hline \multirow{6}{*}{Baseline} & & i-vector & 3.4689 & 3.0395 & 2.9271 \\  & & d-vector & 3.7577 & 2.9342 & 3.1697 \\  & & r-vector & 4.2289 & 3.5635 & 3.5714 * & 3.5306 \\  & & x-vector & 6.3205 & 5.9704 & 6.1229 \\  & 0.027 & i-vector & 3.0575 & 2.9434 & 2.6338 \\  & 0.134 & d-vector & 3.2286 & 2.5765 & 2.6918 \\  & 0.055 & r-vector & 3.5621 & 3.1058 & 3.0593 \\  & 0.025 & x-vector & 5.8561 & 5.509 & 5.7785 \\  & Score fusion & & **2.2855** & **1.7897** & **1.7363** \\ \hline \hline \end{tabular}
\end{table} 
Table 2: EERs (%) of the baseline and proposed methods for each embedding on the validation and evaluation trials of RSR 2015 part 2. (* means the EER of the PCA-PLDA). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{3}{*}{**Method**} & \multirow{3}{*}{**Embedding**} & \multicolumn{4}{c}{**EER (\%)**} \\ \cline{3-5}  & & & **Validation** & **Male** & **Female** \\ \hline \multirow{6}{*}{Baseline} & & i-vector & 3.4689 & 3.0395 & 2.9271 \\  & & d-vector & 3.7577 & 2.9342 & 3.1697 \\  & & r-vector & 4.2289 & 3.5635 \\  & & r-vector & 4.2186 * & 3.5714 * & 3.5306 \\  & & x-vector & 6.3205 & 5.9704 & 6.1229 \\  & 0.027 & i-vector & 3.0575 & 2.9434 & 2.6338 \\  & 0.134 & d-vector & 3.2286 & 2.5765 & 2.6918 \\  & 0.055 & r-vector & 3.5621 & 3.1058 & 3.0593 \\  & 0.025 & x-vector & 5.8561 & 5.509 & 5.7785 \\  & Score fusion & & **2.2855** & **1.7897** & **1.7363** \\ \hline \hline \end{tabular}
\end{table}