Related papers: Manifold learning and optimization using tangent s…

We consider the problem of maximizing the $\ell_1$ norm of a linear map over the sphere, which arises in various machine learning applications such as orthogonal dictionary learning (ODL) and robust subspace recovery (RSR). The problem is…

Optimal transport (OT) has recently found widespread interest in machine learning. It allows to define novel distances between probability measures, which have shown promise in several applications. In this work, we discuss how to…

A flag is a sequence of nested subspaces. Flags are ubiquitous in numerical analysis, arising in finite elements, multigrid, spectral, and pseudospectral methods for numerical PDE; they arise in the form of Krylov subspaces in matrix…

Unsupervised attributed graph representation learning is challenging since both structural and feature information are required to be represented in the latent space. Existing methods concentrate on learning latent representation via…

Modern sample points in many applications no longer comprise real vectors in a real vector space but sample points of much more complex structures, which may be represented as points in a space with a certain underlying geometric structure,…

This paper proposes a novel paradigm for machine learning that moves beyond traditional parameter optimization. Unlike conventional approaches that search for optimal parameters within a fixed geometric space, our core idea is to treat the…

A generative modeling framework is proposed that combines diffusion models and manifold learning to efficiently sample data densities on manifolds. The approach utilizes Diffusion Maps to uncover possible low-dimensional underlying (latent)…

In numerous robotics and mechanical engineering applications, among others, data is often constrained on smooth manifolds due to the presence of rotational degrees of freedom. Common datadriven and learning-based methods such as neural…

Manifold models provide low-dimensional representations that are useful for processing and analyzing data in a transformation-invariant way. In this paper, we study the problem of learning smooth pattern transformation manifolds from image…

Learning meaningful abstract models of Markov Decision Processes (MDPs) is crucial for improving generalization from limited data. In this work, we show how geometric priors can be imposed on the low-dimensional representation manifold of a…

We introduce in this paper a manifold optimization framework that utilizes semi-Riemannian structures on the underlying smooth manifolds. Unlike in Riemannian geometry, where each tangent space is equipped with a positive definite inner…

Symmetric positive definite (SPD) matrices (e.g., covariances, graph Laplacians, etc.) are widely used to model the relationship of spatial or temporal domain. Nevertheless, SPD matrices are theoretically embedded on Riemannian manifolds.…

For covariance-based image descriptors, taking into account the curvature of the corresponding feature space has been shown to improve discrimination performance. This is often done through representing the descriptors as points on…

LiDAR-based place recognition serves as a crucial enabler for long-term autonomy in robotics and autonomous driving systems. Yet, prevailing methodologies relying on handcrafted feature extraction face dual challenges: (1) Inconsistent…

This paper presents a mathematical framework for analyzing machine learning models through the geometry of their induced partitions. By representing partitions as Riemannian simplicial complexes, we capture not only adjacency relationships…

It is proven that encoding images and videos through Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, can lead to increased classification performance. Taking into account manifold…

Choosing the right representation for geometry is crucial for making 3D models compatible with existing applications. Focusing on piecewise-smooth man-made shapes, we propose a new representation that is usable in conventional CAD modeling…

This paper investigates the challenge of learning image manifolds, specifically pose manifolds, of 3D objects using limited training data. It proposes a DNN approach to manifold learning and for predicting images of objects for novel,…

Image super-resolution remains an important research topic to overcome the limitations of physical acquisition systems, and to support the development of high resolution displays. Previous example-based super-resolution approaches mainly…

The techniques and analysis presented in this paper provide new methods to solve optimization problems posed on Riemannian manifolds. A new point of view is offered for the solution of constrained optimization problems. Some classical…