# Autonomous Learning Interactive Features for Hyperspectral Remotely Sensed Data
## 2 Materials and Methods
### Autonomous Learning Based on Sparse Multiclass Logistic Regression  
The core of sparse optimization is to use the L1 norm as the constraint term of the regression problem. Although both the L1 norm and the L2 norm can solve the problem of regression model overfitting, the biggest difference between the two is that the L1 norm can make the part unimportant weight value of the feature becomes 0, so as to achieve the effect of sparse model coefficients. The estimated value of the sparse model with the L1 norm is expressed as follows:  
\[\hat{\alpha}=\text{argmin}\left\{\sum\limits_{i=1}^{n}\left[\left(y_{i}-\sum \limits_{j}\alpha_{j}x_{ij}\right)^{2}+\lambda\sum\limits_{j}|\alpha_{j}| \right]\right\} \tag{5}\]  
Among them, \(n\) is the number of samples, \(x_{ij}\) represents the predictor variable, \(y_{i}\) is the dependent variable of the predictor variable. The smaller the penalty coefficient \(\lambda\), the lower the degree of compression, and the more variables that are not 0; on the contrary, the larger \(\lambda\), the greater the degree of compression, and the fewer variables that are not 0. With the development of the Least Absolute Shrinkage and Selection Operator (LASSO), logistic regression has more fields for applications. [PERSON] extended the Lasso group (gLasso) to logistic regression for high-dimensional problems and generalized linear model to solve the corresponding convex optimization problem [29]. After [PERSON] proposed multiclass logistic regression [30], [PERSON]'s contraction estimation method can be extended to multiclass logistic regression, and the resulting likelihood function of multiclass logistic regression can be written as:  
\[L(y|\alpha)=\prod\limits_{i=1}^{n}\prod\limits_{k=1}^{K}\left(\frac{\exp( \alpha^{(k)}x_{i})}{\sum\limits_{l=1}^{K}\exp(\alpha^{(l)}x_{i})}\right)^{y_{ i}{}^{(k)}} \tag{6}\]  
where \(x_{i}\) is the \(P\)-dimensional feature vector and \(y_{i}\) is the vector of the \(K\)-dimensional category label. When the cross-entropy error function is used for multiclass problems, the goal of maximum likelihood estimation can be achieved by minimizing the negative log-likelihood function:  
\[E(\alpha)=-\ln L(y|\alpha)\]  
\[=-\sum\limits_{i=1}^{n}\left(\sum\limits_{k=1}^{K}y_{i}{}^{(k)}\alpha^{(k)}x_{ i}+\ln\left(\sum\limits_{l=1}^{K}\exp\left(\alpha^{(l)}x_{i}\right)\right)\right) \tag{7}\]  
The sparse representation can eliminate unimportant features from high-dimensional data and leave effective features [31]. Therefore, this paper introduces the L1 norm on the basis of the multiclass logistic regression model, and the parameter estimation is in the following form:  
\[\alpha=\text{argmin}\Bigg{\{}E(\alpha)+\lambda\sum_{j}^{P}\max_{k=1,2,\dots K} \left(\left\|\alpha^{(k)}\right\|\right)\Bigg{\}} \tag{8}\]  
\(\left\|\alpha^{(k)}\right\|\) represents the L1 norm as a penalty term of the likelihood function. In this way, the sparse processing of the parameters is performed, and many components that are not important to the category are set to zero. However, this process completes high-dimensional data classification and feature selection, and the parameters will be estimated by cross-validation in the experiment. The proposed method is described as follows (Algorithm 1). ```
Input: An HSI dataset \(X\) and the corresponding label vectors \(Y\), the shape of \(X\) is \((H\times W)\times C\) and the shape of \(Y\) is \((H\times W)\times 1\) Step1: Normalize the pixel values of all bands of the image between 0 and 1. Step2: Calculate the median value for every 9 bands, the shape of the dataset is changed to \((H\times W)\times C_{1}\), \(C_{1}=\lfloor C/9\rfloor+1\). Then, calculate the interaction between any two bands and express it as \(\frac{X_{j}}{X_{k}}\), which can be seen in equation (4). Step3: Randomly divide the HSI dataset \(X\) into \(X_{1}\) and \(X_{2}\), which represent the training data and testing data. Likewise, \(Y_{1}\) and \(Y_{2}\) are the corresponding label vector.