Related papers: A Lie Group Approach to Riemannian Batch Normaliza…

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…

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…

The Bayesian Learning Rule provides a framework for generic algorithm design but can be difficult to use for three reasons. First, it requires a specific parameterization of exponential family. Second, it uses gradients which can be…

In this study, we investigate stochastic optimization on Riemannian manifolds, focusing on the crucial variance reduction mechanism used in both Euclidean and Riemannian settings. Riemannian variance-reduced methods usually involve a…

Reformulating computer vision problems over Riemannian manifolds has demonstrated superior performance in various computer vision applications. This is because visual data often forms a special structure lying on a lower dimensional space…

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…

Dictionary leaning (DL) and dimensionality reduction (DR) are powerful tools to analyze high-dimensional noisy signals. This paper presents a proposal of a novel Riemannian joint dimensionality reduction and dictionary learning (R-JDRDL) on…

As an indispensable component, Batch Normalization (BN) has successfully improved the training of deep neural networks (DNNs) with mini-batches, by normalizing the distribution of the internal representation for each hidden layer. However,…

Many scientific and geometric problems exhibit general linear symmetries, yet most equivariant neural networks are built for compact groups or simple vector features, limiting their reuse on matrix-valued data such as covariances, inertias,…

Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. The strategies employed to investigate their theoretical properties mainly rely…

We propose a novel deep neural network methodology for density estimation on product Riemannian manifold domains. In our approach, the network directly parameterizes the unknown density function and is trained using a penalized maximum…

Equivariant neural networks incorporate symmetries into their architecture, achieving higher generalization performance. However, constructing equivariant neural networks typically requires prior knowledge of data types and symmetries,…

The renormalization group (RG) is an essential technique in statistical physics and quantum field theory, which considers scale-invariant properties of physical theories and how these theories' parameters change with scaling. Deep learning…

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…

A Lie group as a 4-dimensional pseudo-Riemannian manifold is considered. This manifold is equipped with an almost product structure and a Killing metric in two ways. In the first case Riemannian almost product manifold with nonintegrable…

Manifold learning has been proven to be an effective method for capturing the implicitly intrinsic structure of non-Euclidean data, in which one of the primary challenges is how to maintain the distortion-free (isometry) of the data…

Batch Normalization (BN), a widely-used technique in neural networks, enhances generalization and expedites training by normalizing each mini-batch to the same mean and variance. However, its effectiveness diminishes when confronted with…

In a previous work, we proposed a geometric framework to study a deep neural network, seen as sequence of maps between manifolds, employing singular Riemannian geometry. In this paper, we present an application of this framework, proposing…

Recent work has constructed neural networks that are equivariant to continuous symmetry groups such as 2D and 3D rotations. This is accomplished using explicit Lie group representations to derive the equivariant kernels and nonlinearities.…

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…