## 2 Architecture of the CAPT Framework
### Training Method of Layer 1  
The diagnosing ability of the design is impacted by the multicollinearity problem among the utterance sample predictors; the partial least square (PLS) regression method is therefore applied to train the feature extraction layer of the French phoneme utterance detectors. PLS is a common class of methods for modeling relations between sets of observed variables by means of latent variables. The underlying assumption is that the observed data is generated by a system or process that is driven by a small number of latent (not directly observed or measured) variables. Its goal is to maximize the covariance between the two parts of a paired data set, even though those two parts are in different spaces. That implies that PLS regression can overcome the multicollinearity problem by modeling the relationships between the predictors. In the case of this paper, we train the first layer of the detector to extract the PLS feature of the samples to facilitate the classifying task of the second layer.  
As presented in [32], let \(\mathbf{X}\) and \(\mathbf{Y}\) be two matrices whose rows are the predictor vectors \(\mathbf{x}_{i}\) and their responses \(\mathbf{y}_{i}\) corresponding to the \(i\)-th sample. According to the nonlinear iterative partial least squares algorithm [31; 45], the optimizing problem of PLS regression is to search for some projection directions that maximizes the covariance of the training and response matrices:  
\[\max_{\mathbf{w}_{x},\mathbf{w}_{y}||\mathbf{w}_{x}||=||\mathbf{w}_{y}||=1}\mathbb{C}(\mathbf{w}_ {x},\mathbf{w}_{y})=\max_{\mathbf{w}_{x},\mathbf{w}_{y}||\mathbf{w}_{x}||=||\mathbf{w}_{y}||=1} \frac{1}{m}\mathbf{w}_{x}^{\mathrm{T}}\mathbf{x}^{\mathrm{T}}\mathbf{Y}\mathbf{w}_{y} \tag{4}\]where \(N\) is the number of training samples, \(\mathbf{w}_{x}\) and \(\mathbf{w}_{y}\) are two unit vectors corresponding to the projection directions. The directions that solve (4) are the first singular vectors \(w_{x}=u_{1}\) and \(w_{y}=v_{1}\) of the singular value decomposition of \(\mathbb{C}_{xy}\)  
\[\mathbb{C}_{xy}=U\Sigma V^{\text{T}} \tag{5}\]  
where the value of the covariance is given by the corresponding singular value \(\sigma_{1}\). In this paper we apply the same data projecting strategy through deflation in order to obtain multiple projecting direction.  
The PLS regression algorithm is programmatically described in Algorithm 1. The inner loop computes the first singular value iteratively, which results in \(u_{i}\) converging to the first right singular vector \(Y^{\text{T}}X_{i}\). Next, the deflation of \(X_{i}\) is computed. Finally, the regression coefficients \(\mathbf{\theta}_{b}\) is given by \(\mathbf{W}^{(1)}=\mathbf{\bar{U}}(\mathbf{p}^{\text{T}}\mathbf{\bar{U}})^{-1}\mathbf{C}^{\text{T}}\), where \(\mathbf{C}\) is a matrix with columns \(c_{i}=\mathbf{Y}^{\text{T}}\mathbf{X}_{i}u_{i}/(u_{i}^{\text{T}}\mathbf{X}_{i}^{\text{T}} \mathbf{X}_{i}u_{i})\)[46].  
```
0: training matrix \(\mathbf{X}\), response variables \(\mathbf{Y}\), projection direction number \(k\)
0: regression coefficients \(\mathbf{W}^{(1)}\)
1: initialization
2:for\(i=1,2,\dots,k\)do
3:\(u_{i}\leftarrow\text{first column of }\mathbf{X}^{\text{T}}\mathbf{Y}\)
4:\(u_{i}\gets u_{i}/||u_{i}||\)
5:repeat
6:\(u_{i}\leftarrow\mathbf{X}_{i}^{\text{T}}\mathbf{Y}\mathbf{Y}^{\text{T}}\mathbf{X}_{i}u_{i}\)
7:\(u_{i}\gets u_{i}/||u_{i}||\)
8:until convergence
9:\(p_{i}\leftarrow\mathbf{X}_{i}^{\text{T}}\mathbf{X}_{i}u_{i}/(u_{i}^{\text{T}}\mathbf{X}_{ i}^{\text{T}}\mathbf{X}_{i}u_{i})\)
10:\(c_{i}\leftarrow\mathbf{Y}^{\text{T}}\mathbf{X}_{i}u_{i}/(u_{i}^{\text{T}}\mathbf{X}_{i}^{ \text{T}}\mathbf{X}_{i}u_{i})\)
11:\(\mathbf{X}_{i+1}\leftarrow\mathbf{X}_{i}(I-u_{i}p_{i}^{\text{T}})\)
12:endfor
13:\(\mathbf{W}^{(1)}=\mathbf{\bar{U}}(\mathbf{p}^{\text{T}}\mathbf{\bar{U}})^{-1}\mathbf{C}^{\text{T}}\)
```  
**Algorithm 1** Pseudocode of PLS regression algorithm