Extrinsic Kernel Ridge Regression Classifier for Planar Kendall Shape Space
This work addresses shape classification problems for researchers in statistics and machine learning dealing with non-Euclidean geometry, but it is incremental as it extends existing kernel methods to a specific manifold.
The authors tackled shape classification on the planar Kendall shape space by proposing a new positive definite kernel, the extrinsic Veronese Whitney Gaussian kernel, and implementing a kernel ridge regression classifier, demonstrating promising performance in real data analysis.
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 $Σ_2^k$, 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 $Σ_2^k$, and their promising performances are illustrated through the real data analysis.