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Application of complex extreme learning machine to multiclass classification problems with high dimensionality: A THz spectra classification problem

机译:复杂极限学习机在高维多类分类问题中的应用:太赫兹频谱分类问题

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We extend extreme learning machine (ELM) classifiers to complex Reproducing Kernel Hilbert Spaces (RKHS) where the input/output variables as well as the optimization variables are complex-valued. A new family of classifiers, called complex-valued ELM (CELM) suitable for complex-valued multiple-input-multiple-output processing is introduced. In the proposed method, the associated Lagrangian is computed using induced RKHS kernels, adopting a Wirtinger calculus approach formulated as a constrained optimization problem similarly to the conventional ELM classifier formulation. When training the CELM, the Karush-Khun-Tuker (KKT) theorem is used to solve the dual optimization problem that consists of satisfying simultaneously smallest training error as well as smallest norm of output weights criteria. The proposed formulation also addresses aspects of quaternary classification within a Clifford algebra context. For 2D complex-valued inputs, user-defined complex-coupled hyper-planes divide the classifier input space into four partitions. For 3D complex-valued inputs, the formulation generates three pairs of complex-coupled hyper-planes through orthogonal projections. The six hyper-planes then divide the 3D space into eight partitions. It is shown that the CELM problem formulation is equivalent to solving six real-valued ELM tasks, which are induced by projecting the chosen complex kernel across the different user-defined coordinate planes. A classification example of powdered samples on the basis of their terahertz spectral signatures is used to demonstrate the advantages of the CELM classifiers compared to their SVM counterparts. The proposed classifiers retain the advantages of their ELM counterparts, in that they can perform multiclass classification with lower computational complexity than SVM classifiers. Furthermore, because of their ability to perform classification tasks fast, the proposed formulations are of interest to real-time applications. (C) 2015 Elsevier Inc. All rights reserved.
机译:我们将极限学习机(ELM)分类器扩展到复杂的复制内核希尔伯特空间(RKHS),其中输入/输出变量以及优化变量是复数值。引入了一个新的分类器系列,称为复值ELM(CELM),适用于复值多输入多输出处理。在所提出的方法中,与传统的ELM分类器公式类似,采用归纳为RKHS核的Wirtinger演算方法将相关的拉格朗日数计算为约束优化问题。训练CELM时,Karush-Khun-Tuker(KKT)定理用于解决双重优化问题,该问题包括同时满足最小的训练误差以及最小的输出权重准则。拟议的表述方式还解决了Clifford代数上下文中的四元分类问题。对于2D复数值输入,用户定义的复耦合超平面将分类器输入空间划分为四个分区。对于3D复数值输入,该公式通过正交投影生成三对复耦合超平面。然后,六个超平面将3D空间划分为八个分区。结果表明,CELM问题表述等效于解决六个实值ELM任务,这些任务是通过将选定的复杂内核投影到不同的用户定义坐标平面上而引发的。基于太赫兹光谱特征的粉末样品分类示例用于证明CELM分类器与其SVM同类产品相比的优势。所提出的分类器保留了其ELM同类产品的优势,因为它们可以执行比SVM分类器更低的计算复杂度的多类分类。此外,由于它们具有快速执行分类任务的能力,因此所提出的公式对于实时应用很有意义。 (C)2015 Elsevier Inc.保留所有权利。

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