基于对数欧氏度量学习的概率黎曼空间量化方法

Probabilistic Riemannian quantification method with log-Euclidean metric learning

在许多机器学习应用中,需要分析的数据可能由对称正定矩阵构成,而经典的欧氏机器学习算法处理这种数据的性能较差。针对此问题,提出一种新的基于对数欧氏度量学习的概率黎曼空间量化方法。该方法将对称正定矩阵看做对数欧氏度量下黎曼流形上的点,采用对数欧氏度量学习距离函数将概率学习矢量量化方法从欧氏空间推广到对称正定黎曼空间。在BCI IV 2a脑电数据集上,该方法相较于概率学习矢量量化方法识别正确率提升20%,高于竞赛第一名;并且计算速度快,模型训练及测试时间分别为基于仿射不变度量的同类型算法的1%和10%。在BCI III IIIa和图像数据集ETH-80上也取得了较好的结果。

In many machine learning applications, the data may be symmetric positive definite(SPD) matrices which are not living in Euclidean space.This paper presented a new probabilistic Riemannian space quantization method based on log-Euclidean metric learning.The proposed method extended the Euclidean probabilistic learning vector quantization(PLVQ) method to deal with SPD matrices by treating them as points on the Riemannian manifold of SPD matrices equipped with log-Euclidean metric, through utilizing a parameterized distance function from log-Euclidean metric learning.On the BCI IV 2 a dataset, the proposed method outperformed Euclidean PLVQ by 20% in terms of recognition accuracy.The proposed method also performs better than the first winner of BCI competition IV on this data set.It obtains comparable classification accuracy to PLVQ using affine invariant Riemannian metric, but requires much less computing time, i.e.only needs 1% of the training time, while 10% of the test time.The proposed method also obtains superior performance on the BCI III IIIa and ETH-80 datasets, showing its effectiveness and efficiency.