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

Batch Normalization is an important approach to advancing deep learning since it allows multiple networks to train simultaneously. A problem arises when normalizing along the batch dimension because B.N.'s error increases significantly as…

Deep neural networks (DNNs) on Riemannian manifolds have garnered increasing interest in various applied areas. For instance, DNNs on spherical and hyperbolic manifolds have been designed to solve a wide range of computer vision and nature…

Modern machine learning increasingly leverages the insight that high-dimensional data often lie near low-dimensional, non-linear manifolds, an idea known as the manifold hypothesis. By explicitly modeling the geometric structure of data…

This paper proposes a general framework of Riemannian adaptive optimization methods. The framework encapsulates several stochastic optimization algorithms on Riemannian manifolds and incorporates the mini-batch strategy that is often used…

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…

Euclidean representations distort data with intrinsic non-Euclidean structure. While Riemannian representation learning offers a solution by embedding data onto matching manifolds, it typically relies on an encoder to estimate densities on…

In recent years, skeleton-based action recognition has become a popular 3D classification problem. State-of-the-art methods typically first represent each motion sequence as a high-dimensional trajectory on a Lie group with an additional…

Batch Normalization (BN) is a milestone technique in the development of deep learning, enabling various networks to train. However, normalizing along the batch dimension introduces problems --- BN's error increases rapidly when the batch…

We consider the problem of density estimation on Riemannian manifolds. Density estimation on manifolds has many applications in fluid-mechanics, optics and plasma physics and it appears often when dealing with angular variables (such as…

This paper studies large-scale optimization problems on Riemannian manifolds whose objective function is a finite sum of negative log-probability losses. Such problems arise in various machine learning and signal processing applications. By…

The goal of this paper is to show how different machine learning tools on the Riemannian manifold $\mathcal{P}_d$ of Symmetric Positive Definite (SPD) matrices can be united under a probabilistic framework. For this, we will need several…

In this paper, we formulate a procedure to obtain a generalization of Milnor frames for left-invariant pseudo-Riemannian metrics on a given Lie group. This procedure is an analogue of the recent studies on left-invariant Riemannian metrics,…

In recent years, a variety of normalization methods have been proposed to help train neural networks, such as batch normalization (BN), layer normalization (LN), weight normalization (WN), group normalization (GN), etc. However,…

We introduce a notion of metric on a Lie groupoid, compatible with multiplication, and we study its properties. We show that many families of Lie groupoids admit such metrics, including the important class of proper Lie groupoids. The…

Unregularized deep neural networks (DNNs) can be easily overfit with a limited sample size. We argue that this is mostly due to the disriminative nature of DNNs which directly model the conditional probability (or score) of labels given the…

Nonlinear dimensionality reduction methods provide a valuable means to visualize and interpret high-dimensional data. However, many popular methods can fail dramatically, even on simple two-dimensional manifolds, due to problems such as…

Batch normalization (BN) has become a critical component across diverse deep neural networks. The network with BN is invariant to positively linear re-scale transformation, which makes there exist infinite functionally equivalent networks…

We develop a self-contained theory of log-Euclidean Lie groups: smooth manifolds diffeomorphic to finite-dimensional vector spaces, equipped with the pullback of a constant Euclidean metric. This framework encompasses symmetric…

This study introduces a new normalization layer termed Batch Layer Normalization (BLN) to reduce the problem of internal covariate shift in deep neural network layers. As a combined version of batch and layer normalization, BLN adaptively…

Non-Euclidean constraints are inherent in many kinds of data in computer vision and machine learning, typically as a result of specific invariance requirements that need to be respected during high-level inference. Often, these geometric…