尖峰模型下高维低样本量设置中支持向量机的渐近性质
Asymptotic Properties of Support Vector Machines in High-Dimension, Low-Sample-Size Settings under a Spiked Model
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中文总结 AI 辅助
本文针对尖峰模型下高维低样本量场景，发现现有SVM理论失效，提出尖峰校正SVM（SC-SVM），证明其一致性并验证性能。
中文摘要 AI 辅助
本文研究了尖峰模型下高维低样本量（HDLSS）设置中支持向量机（SVM）的渐近性质。现有的HDLSS背景下SVM理论依赖于HDLSS数据的几何表示，该表示要求协方差矩阵的特征值不占主导地位。我们首先证明在尖峰模型下几何表示不成立。我们证明HDLSS数据的Gram矩阵依分布收敛于一个随机矩阵，即HDLSS数据收敛到有限维空间中的随机配置，其维数由尖峰数量决定。我们证明SVM的误分类率不趋于零，即SVM不具备一致性性质。我们还证明偏差校正SVM（BC-SVM）在此设置下不能给出令人满意的性能，因为偏差项本身需要修改。为了克服这些困难，我们提出了一种尖峰校正SVM（SC-SVM）。我们证明当样本量趋于无穷时，SC-SVM具备一致性性质，并且样本量的增长是至关重要的，因为任何基于投影的方法在样本量固定时都会失败。最后，我们通过数值模拟检验了分类器的性能。
英文摘要
In this paper, we consider asymptotic properties of the support vector machine (SVM) in high-dimension, low-sample-size (HDLSS) settings under a spiked model. The existing theory of the SVM in the HDLSS context relies on the geometric representation of HDLSS data, which requires that the eigenvalues of the covariance matrices are not dominant. We first show that the geometric representation does not hold under the spiked model. We show that the Gram matrix of HDLSS data converges in distribution to a random matrix, namely, the HDLSS data converge to a random configuration in a finite-dimensional space whose dimension is given by the number of the spikes. We show that the misclassification rates of the SVM do not tend to zero, that is, the SVM does not hold the consistency property. We also show that the bias-corrected SVM (BC-SVM) does not give preferable performance in this setting because the bias term itself should be modified. In order to overcome such difficulties, we propose a spike-corrected SVM (SC-SVM). We show that the SC-SVM holds the consistency property when the sample size goes to infinity, and that the growth of the sample size is essential in the sense that any projection-based procedure fails when the sample size is fixed. Finally, we check the performance of the classifiers by numerical simulations.
