## 4 Results and Discussion  
### Parameter Selection of the LRSC Algorithm  
LRSC algorithm requires two parameters while solving the sparse solution. One of these parameters is \(\lambda\) in Equation (9). Similarly to the normalized parameter in ridge regression, this parameter must be considered in the optimization problem. However, when it is actually applied to spectral data, there is no need to emphatically optimize it. According to the literature [29], its cross-validation error can be expressed as:  
\[C(\lambda)=\frac{1}{m}\sum\
olimits_{k=1}^{K}\sum\limits_{i\in F_{k}}\!\! \left(y_{i}-f_{\lambda}^{-k}(x_{i})\right)^{2} \tag{12}\]  
where \(m\) and \(K\) denote the total number of samples and K-fold cross validation respectively, while \(F_{k}\) denotes the \(k\) subset. In addition, \(y_{i}\) and \(f_{\lambda}^{-k}(x_{i})\) denote the one dimensional real value and estimated value modeled by dividing by the \(k\)-th subset, respectively. As can be seen from Equation (12), the selection of \(\lambda\) affects the value of \(C(\lambda)\). Here, the value of \(C(\lambda)\) was the average of the quadratic sum of the error of each variable. However, for the LRSC method, for spectral data with high dimensions, i.e., a large value of \(m\) (up to thousands or even tens of thousands of dimensions), the quadratic sumapproaches a small value. Consequently, the C(\(\lambda\)) is small and selection of \(\lambda\) has little effect on the result. When the LRSC method is applied, the value of \(\lambda\) is \(10^{-3}\), which we will discuss in Section 4.3.  
The other parameter, tolerance \(T\), represents the fault tolerance of the termination of iteration of the BPDN problem. Different from the upper limit of the fault tolerance \(\varepsilon\) in Equation (8), it shows the rate of change of the function \(f\), \(f=\frac{1}{2}\cdot\|y-Ax\|_{2}^{2}+\lambda\|x\|_{1}\). If this rate is greater than \(T\), the tracking process will be terminated. Figure 2 shows the sparse coefficients obtained according to different \(T\) values when the regression coefficients of the prediction samples are solved under a certain division of samples.  
In Figure 2, A, B, C, and D denote the distribution of regression coefficients when \(T\) is \(10^{-1}\), \(10^{-2}\), \(10^{-3}\), and \(10^{-4}\) respectively. The \(x\) axis represents the training samples and the \(y\) axis stands for the regression coefficient value for each of the samples. Each regression coefficient has 100 dimensions, corresponding to the 100 training samples, and, under the condition of known prior class locations (there are in all five classes with 20 samples for each class, separated by dashed lines), the sparse distribution of the regression coefficient determines that the prediction samples are linearly represented by the training samples at the corresponding locations. The figure actually shows the optimization results of the first class of prediction samples under different fault tolerances. As can be seen, selection of different values of \(T\) had little impact on regression coefficients for the spectral data but it affected the sparseness degree of the sparse solution. In fact, for the spectral data, being highly dimensional data, as stated earlier, weakens the influence of this parameter on the performance of the algorithm to some extent. According to the definition of the rate of change of this objective function, the smaller the value of \(T\), the smoother the convergence of the overall objective function. Here, parameter \(T\) was set to be \(10^{-1}\) in LRSC algorithm.