Related papers: Isometric Immersion Learning with Riemannian Geome…
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…
We use a new method to give conditions for the existence of a local isometric immersion of a Riemannian $n$-manifold $M$ in $\mathbb{R}^{n+k}$, for a given $n$ and $k$. These equate to the (local) existence of a $k$-tuple of scalar fields…
A fundamental problem in differential geometry is to characterize intrinsic metrics on a two-dimensional Riemannian manifold ${\mathcal M}^2$ which can be realized as isometric immersions into $\R^3$. This problem can be formulated as…
We revisit the task of learning a Euclidean metric from data. We approach this problem from first principles and formulate it as a surprisingly simple optimization problem. Indeed, our formulation even admits a closed form solution. This…
Riemannian manifolds have been widely employed for video representations in visual classification tasks including video-based face recognition. The success mainly derives from learning a discriminant Riemannian metric which encodes the…
Manifold learning offers nonlinear dimensionality reduction of high-dimensional datasets. In this paper, we bring geometry processing to bear on manifold learning by introducing a new approach based on metric connection for generating a…
This paper is devoted to investigating the isometric immersion problem of Riemannian manifolds in a high codimension. It has recently been demonstrated that any short immersion from an $n$-dimensional smooth compact manifold into…
The isometric immersion of two-dimensional Riemannian manifolds or surfaces in the three-dimensional Euclidean space is a fundamental problem in differential geometry. When the Gauss curvature is negative, the isometric immersion problem is…
Modern generative modeling methods have demonstrated strong performance in learning complex data distributions from clean samples. In many scientific and imaging applications, however, clean samples are unavailable, and only noisy or…
In the domain of image-set based classification, a considerable advance has been made by representing original image sets as covariance matrices which typical lie in a Riemannian manifold. Specifically, it is a Symmetric Positive Definite…
In this paper, we propose RiemannianFlow, a deep generative model that allows robots to learn complex and stable skills evolving on Riemannian manifolds. Examples of Riemannian data in robotics include stiffness (symmetric and positive…
Although Deep Learning (DL) has achieved success in complex Artificial Intelligence (AI) tasks, it suffers from various notorious problems (e.g., feature redundancy, and vanishing or exploding gradients), since updating parameters in…
Riemannian geometry provides the fundamental framework for optimization on nonlinear spaces such as matrix manifolds, which arise in machine learning, signal processing, and robotics. While the underlying theory is classical, existing…
Many of the tools available for robot learning were designed for Euclidean data. However, many applications in robotics involve manifold-valued data. A common example is orientation; this can be represented as a 3-by-3 rotation matrix or a…
Deep generative models learn a mapping from a low dimensional latent space to a high-dimensional data space. Under certain regularity conditions, these models parameterize nonlinear manifolds in the data space. In this paper, we investigate…
Latent variable models are powerful tools for learning low-dimensional manifolds from high-dimensional data. However, when dealing with constrained data such as unit-norm vectors or symmetric positive-definite matrices, existing approaches…
Recent methods in geometric deep learning have introduced various neural networks to operate over data that lie on Riemannian manifolds. Such networks are often necessary to learn well over graphs with a hierarchical structure or to learn…
We present a new technique that enables manifold learning to accurately embed data manifolds that contain holes, without discarding any topological information. Manifold learning aims to embed high dimensional data into a lower dimensional…
In machine learning, there is a long history of trying to build neural networks that can learn from fewer example data by baking in strong geometric priors. However, it is not always clear a priori what geometric constraints are appropriate…
Manifold learning seeks a low dimensional representation that faithfully captures the essence of data. Current methods can successfully learn such representations, but do not provide a meaningful set of operations that are associated with…