Algorithm 4: Procedure for developing dual-level feature extraction and selection.

Estimate a region $\theta$ that has dual optimum ${\theta }^{*}\left(\lambda ,\alpha \right)$ of Equations (10) and (11) for a given pair of parameter values $\left(\lambda ,\alpha \right)$. The two optimization problems are solved as follows: $$s_g^{*}=\underset{{\xi }_g}{\sup }\left\{\Vert S_1\left({\xi }_g\right)\Vert :{\xi }_g\in {\Xi }_g\supseteq X_g^T\mathrm{\Theta }\right\},\mathrm{where}{X}_g^T\mathrm{\Theta }=\left\{X_g^T\theta :\theta \in \mathrm{\Theta }\right\},$$ $$t_{gk}^{*}=\underset{\theta }{\sup }\left\{\left|x_{gk}^T\theta \right|:\theta \in \mathrm{\Theta }\right\},\mathrm{where}{x}_{gk}\mathrm{is}\mathrm{the}{k}^{th}\mathrm{column}\mathrm{of}{X}_g$$ . The dual feature screening ensures the form as: $$s_g^{*}<\alpha \sqrt{n_g}\Rightarrow {\beta }_g^{*}\left(\lambda ,\alpha \right)=0$$ $$t_{gk}^{*}\le 1\Rightarrow {\left[{\beta }_g^{*}\left(\lambda ,\alpha \right)\right]}_k=0,$$ where the optimal solution of SGL in (9) is expressed as $\beta *\left(\lambda ,\alpha \right)$, giving the best extracted and selected features.