## 2 Materials and Methods
### Multinomial Logistic Regression (MLR)  
The set \(\mathrm{c}\equiv\left[\mathrm{c}^{(1)^{T}},\ \mathrm{c}^{(2)^{T}},\ \ldots,\ \mathrm{c}^{(N-1)^{T}}\right]^{T}\) represents regressors (logistic) and \(\mathrm{c}^{(r)}\) is the \(\mathrm{r}\mathrm{r}\mathrm{t}\) class set of regressors [28]. The posterior-class probabilities, as well as regressors, are calculated by the MLR model as given by  
\[p(y_{\mathrm{i}}=r|x_{\mathrm{i}},\ \mathrm{c})=\frac{\exp\left(\mathrm{c}^{(r)} f(x_{\mathrm{i}})\right)}{\sum_{\mathrm{r}=1}^{N}\exp\left(\mathrm{c}^{(r)} f(x_{\mathrm{i}})\right)} \tag{1}\]  
The important point here is that the value of \(\mathrm{c}^{(r)}=0\) because the density is not dependent on translation on the regressor \(\mathrm{c}^{(r)}\). The function \(f(x)\equiv\left[f_{1}(x),\ f_{2}(x),\ \ldots,\ f_{k}(x)\right]^{T}\) where \(k\) represents the total number of feature vectors. The \(f\) (linear/nonlinear) is given by \(f(x_{\mathrm{i}})\equiv\left[1,\ x_{\mathrm{i},1},\ \ldots,\ x_{\mathrm{i}j} \right]^{T}\), where \(x_{\mathrm{i},j}\) represents the jth element of \(x_{\mathrm{i}}\). The kernels are given by \(f(x_{\mathrm{i}})\equiv\left[1,\ F_{x,x_{\mathrm{i},1}},\ \ldots,\ F_{x_{\mathrm{i},N}}\right]^{T}\), here, \(F_{x_{\mathrm{i},N}}:=F(x_{\mathrm{i}},\ x_{\mathrm{j}}):=\left(\zeta(x_{ \mathrm{i}}),\ \zeta(x_{\mathrm{j}})\right)\), and some kind of symmetry also exists in the function F. Parameters are mapped in a nonlinear manner. For this, radial basis function has been employed represented as:  
\[F(x,\ m)\equiv\frac{\exp\left(-||x-m\ ||^{2}\right)}{(2\rho^{2})} \tag{2}\]  
It is important to learn the class densities for regressors' estimation. The class densities are estimated using labeled and unlabeled samples, where \(S_{L}\equiv\{(y_{1},\ x_{1}),\ \ldots,\ (y_{L},\ x_{L})\) represents labeled samples (small set) and \(S_{U}\equiv\{x_{\mathrm{L}+1},\ \ldots,\ x_{\mathrm{L}+U}\}\) represents unlabeled samples (large set). Posterior density using labeled as well as unlabeled data is given by:  
\[p(c|Y_{L},\ D_{L},\ D_{U})\propto p(Y_{L}|D_{L},\ D_{U},\ c)\,p(z|D_{L},\ D_{U}) \equiv p(Y_{L}|D_{L},c)\,p\big{(}c|D_{L+U}\big{)}, \tag{3}\]  
where \(Y_{L}\equiv\{y_{1},\ y_{2},\ \ldots,\ y_{L}\}\) represents set of labels in \(S_{L},\ D_{L}\equiv\{x_{1},\ \ldots,\ x_{L}\}\) represents feature vectors set in \(S_{L}\), and \(D_{L+U}\) represents labeled as well as unlabeled sets of samples. The maximum-a-posteriori (MAP) estimation for regressors is computed as given by \(\mathrm{c}=\operatorname*{argmax}_{\mathrm{c}}\{1(c)+\log p\,(c|X_{L+U})\}\), where  
\[l(c)\equiv\log p(Y_{L}|D_{L},\ c)\equiv\log\prod_{\mathrm{i}=1}^{L}p(y_{ \mathrm{i}}|x_{\mathrm{i},c})\equiv\ \sum_{\mathrm{i}=1}^{L}\,(x_{\mathrm{i}}^{T}\,\mathrm{c}^{(y_{\mathrm{i}})}- \log\sum_{\mathrm{j}=1}^{K}\exp\left(x_{\mathrm{i}}^{T}\mathrm{c}^{(\mathrm{ j})})\right)\,. \tag{4}\]  
Here, \(l(c)\) is the log-likelihood function over \(c\) provided the samples have labels \(S_{L}\) and \(p(c|D_{L+U})\) where the latter works as prior information over \(c\). Further, the Gaussian prior is given by \(p(c|\Gamma)\propto\exp\left\{-\frac{1}{2}c^{T}Tc\right\}\), where \(\Gamma\) is a precision matrix and \(\Gamma=\Gamma(D_{L+U})\) is built in such a way that the regressors \(c\) is promoted by the density \(p(c|\Gamma)\). This leaves samples in the class having similar features \(f(x)\) (either labeled or unlabeled). It is important here to compute the distance between features.