# Regularized Within-Class Precision Matrix Based PLDA in Text-Dependent Speaker Verification
## 7 Conclusions  
In this paper, we improved the conventional PLDA by proposing the GLASSO-PLDA, in which the GLASSO-regularized within-class precision matrix was used to estimate the PLDA model. The GLASSO makes empirical within-class precision matrices sparse. It has the effects of reducing the estimation error in the within-class precision matrices and of reflecting a conditional independence structure on the variables. We assumed that the empirical within-class precision matrices would have large errors due to the limited amount of data and expected that the reduction of the estimation error would lead to performance improvement. From the experimental results on the trials on a public database RSR 2015 parts 1 and 2, we found that the GLASSO-PLDA demonstrated the performance improvement when the within-class covariance/precision matrix of the embedding is close to diagonal. That is, the performance of the PLDA can be improved using GLASSO regularization on the empirical within-class precision matrix when the covariance/precision matrix is close to diagonal. With system fusion, we also have achieved significant performance improvements in the task of TD-SV. The GLASSO-PLDA can be directly applied to the TD-SV systems based on the conventional PLDA without changing the structure of the systems. Therefore, it can be applied to any kinds of applications that uses the TD-SV systems based on the PLDA. In the future, we will apply the GLASSO-PLDA onto noisy condition, where the within-class variability \(\mathbf{\Sigma}_{w}=\mathbf{\Sigma}_{ch}+\mathbf{\Sigma}_{e}+\mathbf{\Sigma}_{noise}\) contains not only both channel variability \(\mathbf{\Sigma}_{ch}\) and residual variability \(\mathbf{\Sigma}_{e}\) but also noise variability \(\mathbf{\Sigma}_{noise}\). Notice that the within-class variability on clean condition, \(\mathbf{\Sigma}_{w}=\mathbf{\Sigma}_{ch}+\mathbf{\Sigma}_{e}\), does not contain \(\mathbf{\Sigma}_{noise}\). In detail, we will evaluate the performance of the GLASSO-PLDA based TD-SV system that is trained using only clean utterances on various noisy conditions, which is the setting close to the environment where real applications are used. This experiment is to confirm whether \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Embedding**} & \multicolumn{4}{c}{**EER (\%)**} \\ \cline{3-5}  & & \multicolumn{2}{c}{**Validation**} & \multicolumn{2}{c}{**Evaluation**} \\ \cline{3-5}  & & & **Male** & **Female** \\ \hline \multirow{4}{*}{Proposed} & i-vector & **3.0575** & **2.9434** & **2.6338** \\  & d-vector & **3.2286** & 2.5765 & 2.6918 \\  & r-vector & **3.5621** & **3.1058** & **3.0593** \\  & x-vector & **5.8561** & **5.509** & **5.7785** \\  & 0 & i-vector & 3.173 & 3.1342 & 2.669 \\ \cline{1-1}  & 0 & d-vector & 3.2393 & **2.5571** & **2.6877** \\ \cline{1-1}  & 0 & r-vector & 3.7074 & 3.1803 & 3.1319 \\ \cline{1-1}  & 9 & x-vector & 5.9019 & 5.5572 & 5.8016 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: EERs (%) of the proposed and banding methods for each embedding on the validation and evaluation trials of RSR 2015 part 2. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multirow{2}{*}{**Embedding**} & \multicolumn{4}{c}{**EER (\%)**} \\ \cline{3-5}  & & & **Validation** & **Male** & **Female** \\ \hline \multirow{4}{*}{Proposed} & i-vector & **2.0842** & **1.2811** & **1.4627** \\  & & d-vector & **1.2036** & **0.9644** & 0.8651 \\  & & r-vector & **1.9761** & **1.5127** & **1.4949** \\  & & x-vector & 3.3587 & 2.6728 & **3.1442** \\  & 0 & i-vector & 2.1904 & 1.3757 & 1.5431 \\ Banding-PLDA with \(k\) of & 0 & d-vector & 1.2398 & 1.0033 & **0.8529** \\  & 0 & r-vector & 2.0544 & 1.5728 & 1.5796 \\  & 11 & x-vector & **3.3551** & **2.6337** & 3.2162 \\ \hline \hline \end{tabular}
\end{table}