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Study on SVM Classifiers for Imbalanced Data Classification Using Quasi-Linear Kernel

Liang Peifeng 早稲田大学

2021.07.29

概要

The classification problem is a key part of machine learning. In real-world, most classi­fication problems are nonlinear and complicated. When building a classification model, the datasets are generally assumed to be balanced and the target is to minimize the loss function to achieve high accuracy and good generalization. However, in most real-world applications, the datasets suffer from imbalance, which results in the learned classifica­tion model to have a decision boundary leaning to the minority class which causes many minority samples misclassified. This will result in poor performance for global classifi­cation, since the minority class which includes much fewer samples than other class is often much more important and valuable. Therefore, simply minimizing loss function to achieve high accuracy is unsuitable to classify the imbalanced datasets. Study on imbalanced data classification has been a very hot research issue in machine learning. In this thesis, support vector machine (SVM) based classifiers with high performance are developed for imbalanced datasets by taking advantages of the quasi-linear SVM.

Quasi-linear SVM is a model based on divide-and-conquer strategy. It uses a multi-local linear model to solve nonlinear classification problems. Unlike SVM with standard k- emel functions which may be considered as black-box models, the quasi-linear SVM first learns information of data structure as prior knowledge and then builds multi-local linear model with interpolation function or piecewise linear model to represent the non­linear classification boundary. From a viewpoint of modeling, the quasi-linear SVM is a two-step modeling method for nonlinear classifications including model building and model optimization. In model building step, the information of data structure is learned and represented as a model in linear regression form. In model optimization step, SVM formulation is applied to optimizing the parameters of model. From the viewpoint of SVM, the quasi-linear SVM is an SVM with a quasi-linear kernel which includes k- emel composition and SVM optimization. In kernel composition step, the learned in­formation is used to compose a kernel function. The second step can be considered as a common kernel SVM for further model optimization. Although the quasi-linear SVM is effective and flexible in classifying nonlinear datasets, it is based on balanced datasets. Simply applying the quasi-linear SVM to imbalanced data classification will not achieve desirable results. Therefore, the goals of this thesis are developing SVM classifiers with high performance for imbalanced datasets of three different scenarios by taking advantages of the quasi-linear SVM:1)a quasi-linear SVM classifier with local offset adjustment for solving within-class imbalance problem; 2) a quasi-linear

SVM classifier with oversampling in feature space to avoid overlapping problems; 3) a quasi-linear SVM classifier for one-class classification with high performance.

The dissertation contains five chapters as follows:

Chapter 1 briefly describes the problems of imbalanced data classification and SVM with quasi-linear kernel. Different problems in imbalanced data classification, such as within-class imbalance problem, overlapping problem caused in SMOTE method and the one-class classification problem are discussed.

Chapter 2 develops a quasi-linear SVM classifier with local offset adjustment for solv­ing within-class imbalance problem by leveraging the multiple local linear models em­bedded in the quasi-linear SVM. Within-class imbalance problems often occur in the imbalanced data classification which worsen the imbalance distribution problem and in­crease the learning complexity. However, most existing methods for imbalanced data ei­ther focus on rectifying the between-class imbalance problem, which is insufficient and inappropriate in many different scenarios. The proposed method tries to solve this prob­lem by developing a simple yet effective SVM classifier with local offset adjustment for imbalance classification problems. First, a geometry-based partitioning method is mod­ified for imbalanced dataset to divide the input space into multiple linearly separable partitions along the potential separation boundary. Then an F-measure SVM is applied to estimate local offsets optimized in each local linear partition. Finally, by constructing a quasi-linear kernel based on the partitioning information, a quasi-linear SVM classi­fier with local offsets is constructed. On 14 real-world datasets, the proposed method outperforms traditional imbalanced classification methods without considering within- class imbalance problem such as weighted SVM (WSVM) and SVM with weighted harmonic mean (WHM) offset (The “win/tie/lose=14/0/0” and “13/0/1” for the F-score and Gmean indexes). Meanwhile, the proposed method has better performance than the previous solutions (local cluster with sampling methods) with the overall evaluation F-score and Gmean: “win/tie/lose= 12/0/2” and “10/0/4” correspondingly.

Chapter 3 develops a quasi-linear SVM classifier with oversampling in multi-linear fea­ture space (MLFS) to solve overlapping problem caused in implementing SMOTE. S- MOTE is a useful and effective oversampling method to balance datasets by creating synthetic minority samples using linear interpolation. However, when dealing with non­linear imbalanced datasets, SMOTE may create wrong samples falling inside majority class which causes the decision boundary to be spread further into majority class. To solve this problem, an MLFS is built based on the quasi-linear kernel which is com­posed from a pretrained neural network. Since data points are separated linearly in the MLFS, implementing SMOTE in the MLFS will not cause overlapping problem. By using the quasi-linear kernel, the proposed oversampling method avoids computing Eu­clidean distances among the samples directly when mapping the samples to the MLFS and oversampling minority class in the MLFS, which makes it easily to be applied to deal with high-dimensional datasets. The proposed method uses unsupervised learning for pretraining neural network, which make it possible to avoid considering the imbal­ance problem at the stage of pretraining the neural network. Finally, in order to imple­ment oversampling minority class fast and effectively, the proposed method implements the SMOTE in kernel level to avoid computing very high dimensional feature vectors directly. On 18 real-world datasets, the proposed method is very competitive and out­performs the traditional SMOTE method on the overall evaluation metrics F-score and Gmean with “win/tie/lose= 18/0/0”,“12/0/6”. Comparing with other previous solutions, the proposed method also achieves better performance with “win/tie/lose= 16/0/2” and “14/0/4” for the F-score and Gmean indexes on the average.

Chapter 4 develops a quasi-linear SVM classifier for one-class classification with high performance by building a piecewise linear model in feature space. The proposed clas­sifier builds a piecewise linear separation boundary in the feature space to separate data from the origin so as to capture a more compact region in the input space. The proposed one-class classifier with more compact region will decrease the probability of outlier objects falling inside the domain of the classifier and then achieve better performance. For this purpose, the input space is first divided into a group of partitions by using a partitioning mechanism of s% winner-take-all autoencoder. A gated linear network is then designed to implement a group of linear classifiers for each partition, in which the gate signals are generated from the autoencoder. By applying a one-class SVM formu­lation to optimize the parameter set of gated linear network, the one-class classifier is implemented in an exact same way as a standard one-class SVM with a quasi-linear kernel composed by using a base kernel with the gate signals. Numerical experiments on various real-world datasets demonstrate the effectiveness of the proposed method. The classification results of the proposed one-class classifier are 6twin/lose=13/l55 and “14/0” for F-score and accuracy indexes compare to traditional one-class SVM on 14 real-world datasets.

Chapter 5 concludes the dissertation. In this thesis, by taking advantage of the quasi- linear SVM, three SVM classifiers with high performance are developed for imbalanced data classification problems: a quasi-linear SVM classifier with local offset adjustment for solve within-class imbalance, a quasi-linear SVM classifier with oversampling in feature space to avoid overlapping problems and a quasi-linear SVM classifier for one- class classification with high performance. Numerical simulation results demonstrated the effectiveness of the proposed SVM classifiers for imbalanced data classifications.

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