Probabilistic Learning Vector Quantization on Manifold of Symmetric Positive Definite Matrices
This addresses classification challenges for manifold-valued data in fields like neuroimaging and computer vision, though it is an incremental adaptation of an existing method to a specific domain.
The authors tackled classification of data represented as symmetric positive definite matrices on a Riemannian manifold by generalizing probabilistic learning vector quantization to this non-Euclidean setting, resulting in a method that demonstrated superior performance on synthetic, image, and EEG data.
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 EEG data demonstrate the superior performance of the proposed method.