The ν support vector machine (ν-SVM) is an enhanced algorithm derived from support vector machines using parameter ν to replace the original penalty coefficient C. Because of the narrower range of ν compared with the infinite range of C, ν-SVM generally outperforms the standard SVM. Granular ball computing is an information fusion method that enhances system robustness and reduces uncertainty. To further improve the efficiency and robustness of support vector algorithms, this paper introduces the concept of multigranularity granular balls and proposes the controllable multigranularity SVM (Con-MGSVM) and the controllable multigranularity support vector regression machine (Con-MGSVR). These models use granular computing theory, replacing original fine-grained points with coarse-grained “granular balls” as inputs to a classifier or regressor. By introducing control parameter ν, the number of support granular balls can be further reduced, thereby enhancing computational efficiency and improving robustness and interpretability. Furthermore, this paper derives and solves the dual models of Con-MGSVM and Con-MGSVR and conducts a comparative study on the relationship between the granular ball SVM (GBSVM) and the Con-MGSVM model, elucidating the importance of control parameters. Experimental results demonstrate that Con-MGSVM and Con-MGSVR not only improve accuracy and fitting performance but also effectively reduce the number of support granular balls.

中文翻译：

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[公式省略]-MGSVM：用于分类和回归的可控多粒度支持向量算法


ν 支持向量机 （ν-SVM） 是从支持向量机派生而来的增强算法，使用参数 ν 替换原始惩罚系数 C。由于 ν 的范围比 C 的无限范围更窄，因此 ν-SVM 的性能通常优于标准 SVM。Granular ball computing 是一种信息融合方法，可增强系统稳健性并减少不确定性。为了进一步提高支持向量算法的效率和鲁棒性，该文引入了多粒度粒球的概念，提出了可控多粒度SVM（Con-MGSVM）和可控多粒度支持向量回归机（Con-MGSVR）。这些模型使用粒度计算理论，将原始的细粒度点替换为粗粒度的“粒度球”，作为分类器或回归器的输入。通过引入控制参数 ν，可以进一步减少支持粒球的数量，从而提高计算效率并提高鲁棒性和可解释性。此外，该文推导并求解了 Con-MGSVM 和 Con-MGSVR 的对偶模型，并对颗粒球 SVM （GBSVM） 和 Con-MGSVM 模型之间的关系进行了比较研究，阐明了控制参数的重要性。实验结果表明，Con-MGSVM 和 Con-MGSVR 不仅提高了精度和拟合性能，而且有效减少了支撑粒球的数量。