Probabilistic Learning Vector Quantization on Manifold of Symmetric Positive Definite Matrices

TL;DR

This paper introduces a probabilistic learning vector quantization method tailored for classifying data on the Riemannian manifold of symmetric positive definite matrices, addressing non-Euclidean geometry challenges.

Contribution

It generalizes the probabilistic learning vector quantization algorithm to Riemannian manifolds, specifically for symmetric positive definite matrices, using Riemannian gradient descent.

Findings

Superior performance on synthetic, image, and EEG data

Effective handling of non-Euclidean geometry in classification

Demonstrates advantages over traditional Euclidean methods

Abstract

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.