Extrinsic Kernel Ridge Regression Classifier for Planar Kendall Shape Space

TL;DR

This paper introduces a new kernel-based classifier for shape data on planar Kendall shape space, extending Gaussian kernels to embedded manifolds and demonstrating promising results in shape classification tasks.

Contribution

It proposes an extrinsic Veronese Whitney Gaussian kernel for Kendall shape space and applies kernel ridge regression for shape classification, extending kernel methods to non-Euclidean manifolds.

Findings

Effective shape classification on Kendall shape space

Extension of Gaussian kernels to embedded manifolds

Promising results in real data analysis

Abstract

Kernel methods have had great success in Statistics and Machine Learning. Despite their growing popularity, however, less effort has been drawn towards developing kernel based classification methods on Riemannian manifolds due to difficulty in dealing with non-Euclidean geometry. In this paper, motivated by the extrinsic framework of manifold-valued data analysis, we propose a new positive definite kernel on planar Kendall shape space $\Sigma_2^k$, called extrinsic Veronese Whitney Gaussian kernel. We show that our approach can be extended to develop Gaussian kernels on any embedded manifold. Furthermore, kernel ridge regression classifier (KRRC) is implemented to address the shape classification problem on $\Sigma_2^k$, and their promising performances are illustrated through the real data analysis.