## 2 Methodology
### Multinomial Logistic Regression via Splitting and Augmented Lagrangian (MLR-LORSAL)  
An NP classifier modeled using \(X_{T}\) is tested on \(X_{V}\), outputs a matrix \(\mu\) of \(m\times C\) dimensions whose entries correspond to NP outputs (i.e., least-square estimation) of a classifier. Let \(u_{ij}\) represent the membership of \(i\)th sample for \(j\)th class. Various methods are proposed for the transformation of least square estimation into the posteriori probabilities [36; 38]. These methods are expensive in terms of computations due to two main aspects: first, in order to estimate \(y_{i}\), they need to approximate the posterior distribution in the case of a Bayesian framework. Secondly, they restrict the training output between 0 and 1. Such methods compute the approximated posterior probabilities in the form of least squares regression i.e., \(f(x_{i},\mu_{j})=p(\mu_{j},x_{i})\)[40]. An alternate to the least squares approach which outputs the probabilities in \([0-1]\) range is MLR which is computed as [40; 41]:  
\[p(y_{i})^{k}=1|(x_{i},w)=\frac{exp(w^{k}h(x_{i}))}{\sum_{k=1}^{K}exp(w^{k}h(x_ {i}))} \tag{1}\]  
where \(h(x)=[h_{1}(x),\dots,h_{L}(x)]^{T}\), mostly termed as features, is a vector of \(L\) fixed functions of the input, \(w=[w^{1},\dots,w^{L-1}]^{T}\). Since the density function represented in Equation (1) is independent of translations of the regressors \(w^{L}\), we take \(w^{L}=0\). Therefore, the posteriori probabilities are computed by using LORSAL algorithm as similar to the work [37]. \(h(x_{i})=[1,\mu_{(x,x_{1})},\dots,\mu_{(x,x_{L})}]\), where \(\mu_{(x_{i},x_{j})}=\mu(x_{i},x_{j})\) are symmetric kernel functions. RBF has been widely used for HSI classification as it improves the data separability in the transformed space [41; 42]. \[\mu(x_{i},x_{j})=\frac{-exp(\|x_{i}-x_{j}\|^{2})}{2\sigma^{2}} \tag{2}\]  
From RBF output, a membership matrix \(\mu_{ij}\) is obtained, which must satisfy the properties [6; 25]\(\sum_{j=1}^{C}\mu_{ij}=1\) and \(0<\sum_{i=1}^{N}\mu_{ij}<1\) where \(\mu_{ij}\in[0,1]\) and \(\mu_{ij}=\mu_{j}(x_{i})\) represents the membership of sample \(i\) for the \(j\)th class [25]. If the approximated posterior probability is close to 1, it represents the true class, whereas wrong class if the probability is close to 0. AL techniques do not need exact probabilities, but they require a ranking of approximated probabilities which makes it easier to calculate the fuzziness [6]. The fuzziness \(F(\mu_{ij})\) from the membership matrix \(\mu_{ij}\) can be expressed as in Equation (3) and it should satisfy the properties defined below [6; 43]. \[F(\mu_{ij})=\frac{-1}{m\times C}\sum_{i=1}^{m}\sum_{j=1}^{C}[\mu_{ij}\log\mu_ {ij}+(1-\mu_{ij})\log(1-\mu_{ij}))] \tag{3}\]  
1. \(F(\mu_{ij})=0\) iff \(\mu_{ij}\) is a crisp set. 2. \(F(\mu_{ij})\) is maximum iff \(\mu_{i}=0.5\). 3. If \(\mu_{ij}<<\sigma_{ij}\) then \(F(\mu_{ij})>>F(\sigma_{ij})\) where \(\mu_{ij}<<\sigma_{ij}\iff min(0.5,\mu(i))\geq min(0.5,\sigma(i))\) and \(min(0.5,\mu(i))\leq max(0.5,\sigma(i))\)
4. \(F(\mu_{ij}\cup\sigma_{ij})+F(\mu_{ij}\cap\sigma_{ij})=F(\mu_{ij})+F(\sigma_{ ij})\)  
where \(\mu_{ij}\) and \(\sigma_{ij}\) are fuzzy sets. We first associate the \(F(\mu)\) with actual and predicted class labels and validation set \(X_{V}\). Then this set is sorted in a descending manner according to the \(F(\mu)\) values. After that \(\hat{m}\) number of misclassified samples having higher fuzziness (i.e., \(F(\mu)>>0.7\) and \(\hat{m}\ll m\)) values are heuristically selected.