Abstract

The recently proposed twin extreme learning machine (TELM) requires solving two quadratic programming problems (QPPs) in order to find two non-parallel hypersurfaces in the feature that brings in the additional requirement of external optimization toolbox such as MOSEK. In this paper, we propose implicit Lagrangian TELM for classification via unconstrained convex minimization problem (ULTELMC) and further suggest iterative convergent schemes which eliminates the requirement of external optimization toolbox generally required in solving the quadratic programming problems (QPPs) of TELM. The solutions to the dual variables of the proposed ULTELMC are obtained using iterative schemes containing ‘plus’ function which is not differentiable. To overcome this shortcoming, the generalized derivative approach and smooth approximation approaches are suggested. Further, to test the performance of the proposed approaches, classification performances are compared with support vector machine (SVM), twin support vector machine (TWSVM), extreme learning machine (ELM), twin extreme learning machine (TELM) and Lagrangian extreme learning machine (LELM). Moreover, non-requirement to solve QPPs makes the iterative schemes find the solution faster as compared to the reported methods that finds the solution in dual space. Computational times required in finding the solutions are also presented for comparison.

中文翻译：

---

基于无约束凸最小化的隐式拉格朗日孪生极限学习机（ULTELMC）

摘要

最近提出的孪生极限学习机（TELM）需要解决两个二次规划问题（QPP），以便在特征中找到两个非平行的超曲面，从而带来了外部优化工具箱（例如MOSEK）的额外要求。在本文中，我们提出了通过无约束凸最小化问题（ULTELMC）进行分类的隐式拉格朗日TELM，并进一步提出了迭代收敛方案，从而消除了解决TELM二次规划问题（QPPs）时通常需要的外部优化工具箱。建议的ULTELMC对偶变量的解决方案是使用包含“加”函数的迭代方案获得的，该方案不可微分。为了克服这个缺点，提出了广义导数法和平滑逼近法。此外，为了测试所提出方法的性能，将分类性能与支持向量机（SVM），双支持向量机（TWSVM），极限学习机（ELM），双极限学习机（TELM）和拉格朗日极限学习机进行比较（LELM）。而且，与报道的在双空间中找到解决方案的方法相比，不需要解决QPP使得迭代方案更快地找到解决方案。还提供了寻找解决方案所需的计算时间以进行比较。双胞胎极限学习机（TELM）和拉格朗日极限学习机（LELM）。此外，与报道的在双空间中找到解决方案的方法相比，不需要解决QPP使得迭代方案更快地找到解决方案。还提供了寻找解决方案所需的计算时间以进行比较。双胞胎极限学习机（TELM）和拉格朗日极限学习机（LELM）。此外，与报道的在双空间中找到解决方案的方法相比，不需要解决QPP使得迭代方案更快地找到解决方案。还提供了寻找解决方案所需的计算时间以进行比较。