State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, 110016, China; Institutes for Robotics and Intelligent Manufacturing, Chinese Academy of Sciences, Shenyang, 110169, China.
State Key Laboratory of Robotics, Shenyang Institute of Automation, Chinese Academy of Sciences, Shenyang, 110016, China; Institutes for Robotics and Intelligent Manufacturing, Chinese Academy of Sciences, Shenyang, 110169, China; University of Chinese Academy of Sciences, Beijing, 100049, China.
Neural Netw. 2021 Oct;142:105-118. doi: 10.1016/j.neunet.2021.04.024. Epub 2021 Apr 28.
In this paper, we develop a new classification method for manifold-valued data in the framework of probabilistic learning vector quantization. In many classification scenarios, the data can be naturally represented by symmetric positive definite matrices, which are inherently points that live on a curved Riemannian manifold. Due to the non-Euclidean geometry of Riemannian manifolds, traditional Euclidean machine learning algorithms yield poor results on such data. In this paper, we generalize the probabilistic learning vector quantization algorithm for data points living on the manifold of symmetric positive definite matrices equipped with Riemannian natural metric (affine-invariant metric). By exploiting the induced Riemannian distance, we derive the probabilistic learning Riemannian space quantization algorithm, obtaining the learning rule through Riemannian gradient descent. Empirical investigations on synthetic data, image data , and motor imagery electroencephalogram (EEG) data demonstrate the superior performance of the proposed method.
在本文中，我们在概率学习向量量化框架中为流形值数据开发了一种新的分类方法。在许多分类场景中，数据可以自然地表示为对称正定矩阵，这些矩阵本质上是生活在弯曲黎曼流形上的点。由于黎曼流形的非欧几里得几何性质，传统的欧几里得机器学习算法在这类数据上的效果不佳。在本文中，我们将概率学习向量量化算法推广到配备黎曼自然度量（仿射不变度量）的对称正定矩阵流形上的数据点。通过利用诱导的黎曼距离，我们推导出概率学习黎曼空间量化算法，通过黎曼梯度下降获得学习规则。对合成数据、图像数据和运动想象脑电图（EEG）数据的实证研究表明了该方法的优越性能。
