This paper presents approaches to compute sparse solutions of Generalized Singular Value Problem (GSVP). The GSVP is regularized by $\ell_1$-norm and $\ell_q$-penalty for $0<q<1$, resulting in the $\ell_1$-GSVP and $\ell_q$-GSVP formulations. The solutions of these problems are determined by applying the proximal gradient descent algorithm with a fixed step size. The inherent sparsity levels within the computed solutions are exploited for feature selection, and subsequently, binary classification with non-parallel Support Vector Machines (SVM). For our feature selection task, SVM is integrated into the $\ell_1$-GSVP and $\ell_q$-GSVP frameworks to derive the $\ell_1$-GSVPSVM and $\ell_q$-GSVPSVM variants. Machine learning applications to cancer detection are considered. We remarkably report near-to-perfect balanced accuracy across breast and ovarian cancer datasets using a few selected features.

翻译：本文提出了计算广义奇异值问题稀疏解的若干方法。通过引入$\ell_1$范数与$0<q<1$条件下的$\ell_q$惩罚项对广义奇异值问题进行正则化，形成了$\ell_1$-GSVP与$\ell_q$-GSVP两种数学模型。采用固定步长的近端梯度下降算法求解这些问题。计算所得解中固有的稀疏性被用于特征选择，并进一步结合非平行支持向量机进行二分类任务。在特征选择过程中，将支持向量机整合至$\ell_1$-GSVP与$\ell_q$-GSVP框架，从而衍生出$\ell_1$-GSVPSVM与$\ell_q$-GSVPSVM两种变体模型。研究探讨了该方法在癌症检测中的机器学习应用。实验结果表明，在乳腺癌和卵巢癌数据集上，仅使用少量筛选特征即可获得接近完美的平衡准确率。