Abstract In this chapter, we collect some information on various constructions of manifolds, orbifolds, and their covers. Notably, discuss the notions fiber product (in sense Thurston) compositum manifolds over a common developable orbifold, difference with set-theoretic product, connected components Galois covers, normal closure cover, relation between commensurability, arithmeticity, existenc...

We propose a simple method to identify continuous Lie algebra symmetry in dataset through regression by an artificial neural network. Our proposal takes advantage of the $\mathcal{O}({\ensuremath{\epsilon}}^{2})$ scaling output variable under infinitesimal transformations on input variables. As are generated post-training, methodology does not rely sampling full representation space or binning ...

In this paper, a novel deep manifold-to-manifold transforming network (DMT-Net) is proposed for action recognition, in which symmetric positive definite (SPD) matrix is adopted to describe the spatial-temporal information of action feature vectors. Since each SPD matrix is a point of the Riemannian manifold space, the proposed DMT-Net aims to learn more discriminative feature by hierarchically ...

The success of semi-supervised manifold learning is highly dependent on the quality of the labeled samples. Active manifold learning aims to select and label representative landmarks on a manifold from a given set of samples to improve semi-supervised manifold learning. In this paper, we propose a novel active manifold learning method based on a unified framework of manifold landmarking. In par...

Dimensionality reduction is required to produce visualizations of high dimensional data. In this framework, one of the most straightforward approaches to visualising high dimensional data is based on reducing complexity and applying linear projections while tumbling the projection axes in a defined sequence which generates a Grand Tour of the data. We propose using smooth nonlinear topographic ...

The goal of Manifold Learning (ML) is to find a description of low-dimensional structure of an unknown q-dimensional manifold embedded in high-dimensional ambient Euclidean space R p , q < p, from their finite samples. There are a variety of formulations of the problem. The methods of Manifold Approximation (MA) reconstruct (estimate) the manifold but don’t find a low-dimensional parameterizati...