Related papers: Riemannian geometry as a unifying theory for robot…
It is shown that Electromagnetism creates geometry different from Riemannian geometry. General geometry including Riemannian geometry as a special case is constructed. It is proven that the most simplest special case of General Geometry is…
Autonomous exploration of unknown environments using a team of mobile robots demands distributed perception and planning strategies to enable efficient and scalable performance. Ideally, each robot should update its map and plan its motion…
Meta-learning, or "learning to learn," aims to enable models to quickly adapt to new tasks with minimal data. While traditional methods like Model-Agnostic Meta-Learning (MAML) optimize parameters in Euclidean space, they often struggle to…
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…
Recent advances in the theory of metric measures spaces on the one hand, and of sub-Riemannian ones on the other hand, suggest the possibility of a "great unification" of Riemannian and sub-Riemannian geometries in a comprehensive framework…
We explore the use of tools from Riemannian geometry for the analysis of symmetric positive definite matrices (SPD). An SPD matrix is a versatile data representation that is commonly used in chemical engineering (e.g.,…
Physical theories grounded in mathematical symmetries are an essential component of our understanding of a wide range of properties of the universe. Similarly, in the domain of machine learning, an awareness of symmetries such as rotation…
This paper presents a Riemannian metric-based model to solve the optimal path planning problem on two-dimensional smooth submanifolds in high-dimensional space. Our model is based on constructing a new Riemannian metric on a two-dimensional…
We introduce Riemannian Flow Matching Policies (RFMP), a novel model for learning and synthesizing robot visuomotor policies. RFMP leverages the efficient training and inference capabilities of flow matching methods. By design, RFMP…
The mathematical theory underlying Hamiltonian mechanics is called symplectic geometry. So symplectic geometry arose from the roots of mechanics and is seen as one of the most valuable links between physics and mathematics today. Symplectic…
This paper describes the pragmatic design and construction of geometric fabrics for shaping a robot's task-independent nominal behavior, capturing behavioral components such as obstacle avoidance, joint limit avoidance, redundancy…
We study the geometrical properties of the utility space (the space of expected utilities over a finite set of options), which is commonly used to model the preferences of an agent in a situation of uncertainty. We focus on the case where…
In many robot motion planning problems, task objectives and physical constraints induce non-Euclidean geometry on the configuration space, yet many planners operate using Euclidean distances that ignore this structure. We address the…
Data-driven Riemannian geometry has emerged as a powerful tool for interpretable representation learning, offering improved efficiency in downstream tasks. Moving forward, it is crucial to balance cheap manifold mappings with efficient…
Learning in Riemannian orbifolds is motivated by existing machine learning algorithms that directly operate on finite combinatorial structures such as point patterns, trees, and graphs. These methods, however, lack statistical…
A complete understanding of the widely used over-parameterized deep networks is a key step for AI. In this work we try to give a geometric picture of over-parameterized deep networks using our geometrization scheme. We show that the…
Navigation is one of the most heavily studied problems in robotics, and is conventionally approached as a geometric mapping and planning problem. However, real-world navigation presents a complex set of physical challenges that defies…
The newest model for space-time is based on sub-Riemannian geometry. In this paper, we use a combination of Lorentzian and sub-Riemannian geometry, the suggest a new model which likes to its ancestors, but with the most efficient in…
Learning from demonstration (LfD) is considered as an efficient way to transfer skills from humans to robots. Traditionally, LfD has been used to transfer Cartesian and joint positions and forces from human demonstrations. The traditional…
High-dimensional data with intrinsic low-dimensional structure is ubiquitous in machine learning and data science. While various approaches allow one to learn a data manifold with a Riemannian structure from finite samples, performing…