## 3 Proposed UDS-Based SSI  
### Joint Minimization Using Iterative Learning  
A block diagram of the proposed SSI method is shown in Figure 3. The major difference from the baseline SSI shown in Figure 1 is that the objective function to be minimized is given by not only the MSE but also the perceptual distance. In this study, the magnitude spectrum, the HE filter coefficients, and the pitch period were predicted from the features derived from the UDS. To estimate the pitch period, we adopted the approach explained in the previous section. Let \(\mathcal{F}_{Y}\) and \(\mathcal{F}_{\Gamma}\) denote the prediction rules for the magnitude spectrum \(\mathbf{Y}\) and the HE filter coefficients \(\mathbf{\Gamma}\), respectively, and then the optimal prediction rules for the MSE-based methods, \(\mathcal{F}_{Y}^{*}\), \(\mathcal{F}_{\Gamma}^{*}\) are given by  
\[\mathcal{F}_{Y}^{*} = \arg\min_{\mathcal{F}_{Y}}D_{MSE}(\mathbf{Y},\mathbf{\hat{Y}}),\] \[\mathcal{F}_{\Gamma}^{*} = \arg\min_{\mathcal{F}_{\Gamma}}D_{MSE}(\mathbf{\Gamma},\mathbf{ \hat{\Gamma}}) \tag{9}\]  
where \(\mathbf{\hat{Y}}=\mathcal{F}_{Y}(\mathbf{X})\) and \(\mathbf{\hat{\Gamma}}=\mathcal{F}_{\Gamma}(\mathbf{X})\). \(\mathbf{X}=\{\mathbf{x}_{n}\}_{n=1}^{N}\) is the set of the UDS features where \(N\) is the total number of the features for constructing the prediction rules. \(\mathbf{Y}\) and \(\mathbf{\Gamma}\) are the sets of the magnitude spectra and the HE filter coefficients, respectively, which are obtained from the speech signals. \(D_{MSE}(X,Y)\) denotes the MSE between \(X\) and \(Y\). Hence, Equation (9) indicates that the prediction rules for two parameters are independently obtained by minimizing each MSE. In the proposed method, construction of the two prediction rules was achieved by jointly minimizing a unique distance measurement as shown in Figure 3.  
\[\mathcal{F}_{Y}^{*},\ \mathcal{F}_{Y}^{*}=\arg\min_{\mathcal{F}_{Y},\ \mathcal{F}_{ \Gamma}}D_{PD}(\mathbf{Y},\mathbf{\hat{Y}},\mathbf{\Gamma},\mathbf{\hat{\Gamma }}) \tag{10}\]  
where \(D_{PD}\) is the perceptual distance measure that was explained in the subsequent section. Simultaneous minimization of \(\mathcal{F}_{Y}^{*}\) and \(\mathcal{F}_{\Gamma}^{*}\) cannot be achieved with a closed-form solution, \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Modality** & **FPE \(\mu\) [Hz]** & **FPR \(\sigma\) [Hz]** & **GPE Rate (\%)** \\ \hline Vision & 48.82 & 39.95 & 73.6 \\ EMG & 60.19 & 34.13 & 88.5 \\ e UDS & 27.39 & 39.33 & 45.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Pitch estimation accuracy for each modality. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Modality**} & \multicolumn{2}{c}{**Voiced Frames (\%)**} & \multicolumn{3}{c}{**Unvoiced Frames (\%)**} \\ \cline{2-7}  & **Precision** & **Recall** & **Accuracy** & **Precision** & **Recall** & **Accuracy** \\ \hline Vision & 84.7 & 95.4 & 90.1 & 74.6 & 43.7 & 59.2 \\ EMG & 78.8 & 99.9 & 89.4 & 17.0 & 0.01 & 8.5 \\ UDS & 87.7 & 88.5 & 88.1 & 76.4 & 74.9 & 75.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Voiced/unvoiced detection accuracy for each modality. and an iterative method was employed in this study, wherein each prediction rule was iteratively updated to minimize the objective function. Beginning with the initial rules \(\mathcal{F}_{Y}^{(0)}\) and \(\mathcal{F}_{\Gamma}^{(0)}\), the conversion rules for each parameter at the \(n\)-th iteration are given by  
\[\mathcal{F}_{Y}^{(i)} = \arg\min_{\mathcal{F}_{Y}}D_{PD}\big{\{}\mathbf{Y},\mathcal{F}_{Y} (\mathbf{X}),\mathbf{\Gamma},\mathcal{F}_{\Gamma}^{(i-1)}(\mathbf{X})\big{\}}\] \[\mathcal{F}_{\Gamma}^{(i)} = \arg\min_{\mathcal{F}_{\Gamma}}D_{PD}\big{\{}\mathbf{Y},\mathcal{F }_{Y}^{(i)}(\mathbf{X}),\mathbf{\Gamma},\mathcal{F}_{\Gamma}(\mathbf{X})\big{\}} \tag{11}\]  
The process is repeated until some convergence threshold is reached.