Abstract. We introduce a mathematical framework for the representation and classification of 3D human actions from sequences of skeletal joint configurations. Each skeleton is modeled as a shape in Kendall’s shape space of 3D landmarks and is embedded into the space of symmetric positive semidefinite matrices with trace equal to one. This extrinsic formulation endows the representation with a linear structure that enables an embedding into a reproducing kernel Hilbert space (RKHS), thereby facilitating the direct use of kernel-based machine learning methods for action classification. Within this framework, sequences of skeletons are represented as smooth parameterized curves in the embedding space, allowing for interpolation and normalization of the number of frames while preserving the geometric structure of the shapes. The linear geometry induced by the embedding simplifies the application of modern learning tools, avoiding the computational and conceptual difficulties associated with intrinsic Riemannian approaches on Kendall’s shape space. To compare action trajectories while accounting for temporal variability, we employ Dynamic Time Warping (DTW) and the global alignment kernel (GAK), which provides a valid positive-definite kernel suitable for classifiers such as support vector machines. Experiments on benchmark skeleton-based human action datasets demonstrate that the proposed approach achieves competitive performance while offering a principled and geometrically grounded representation of 3D action dynamics.

We introduce a novel exploratory technique, termed biarchetype analysis, which extends archetype analysis to simultaneously identify archetypes of both observations and features. This innovative unsupervised machine learning tool aims to represent observations and features through instances of pure types, or biarchetypes, which are easily interpretable as they embody mixtures of observations and features. Furthermore, the observations and features are expressed as mixtures of the biarchetypes, which makes the structure of the data easier to understand. We propose an algorithm to solve biarchetype analysis. Although clustering is not the primary aim of this technique, biarchetype analysis is demonstrated to offer significant advantages over biclustering methods, particularly in terms of interpretability. This is attributed to biarchetypes being extreme instances, in contrast to the centroids produced by biclustering, which inherently enhances human comprehension. The application of biarchetype analysis across various machine learning challenges underscores its value.

In this paper, we develop an approach to exploit kernel methods with data lying on the m-D Kendall shape space. When data arise in a finite-dimensional curved Riemannian manifold, as in this case, the usual Euclidean computer vision and machine learning algorithms must be treated carefully. A good approach is to use positive definite kernels on manifolds to embed the manifold with its corresponding metric in a high-dimensional reproducing kernel Hilbert space, where it is possible to utilize algorithms developed for linear spaces. Different Gaussian kernels can be found in the literature on the 2-D Kendall shape space to perform this embedding. The main novelty of this work is to provide a Gaussian kernel for the m-D Kendall shape space. This new Kernel coincides in the case m=2 with the Gaussian kernels most widely used in the Kendall planar shape space and allows to define an embedding of the m-D Kendall shape space into a reproducible kernel Hilbert space. As far as we know, the complexity of the m-D Kendall shape space has meant that this embedding has not been addressed in the literature until now. This methodology will be tested on a machine learning problem with a simulated and a real data set.