Related papers: Finding geodesics with the Deep Ritz method
Nowadays the geometric approach in optics is often used to find out media parameters based on propagation paths of the rays because in this case it is a direct problem. However inverse problem in the framework of geometrical optics is…
Consumer electronic devices such as mobile handsets, goods tagged with RFID labels, location and position sensors are continuously generating a vast amount of location enriched data called geospatial data. Conventionally such geospatial…
Geodesics escape is widely used to study the scattering of hyperbolic equations. However, there are few progresses except in a simply connected complete Riemannian manifold with nonpositive curvature. We propose a kind of complete…
Existing dimensionality reduction methods are adept at revealing hidden underlying manifolds arising from high-dimensional data and thereby producing a low-dimensional representation. However, the smoothness of the manifolds produced by…
Recent years have witnessed a growth in mathematics for deep learning--which seeks a deeper understanding of the concepts of deep learning with mathematics and explores how to make it more robust--and deep learning for mathematics, where…
Districting is a complex combinatorial problem that consists in partitioning a geographical area into small districts. In logistics, it is a major strategic decision determining operating costs for several years. Solving districting…
We give new results for problems in computational and statistical machine learning using tools from high-dimensional geometry and probability. We break up our treatment into two parts. In Part I, we focus on computational considerations in…
For a set of robots (or agents) moving in a graph, two properties are highly desirable: confidentiality (i.e., a message between two agents must not pass through any intermediate agent) and efficiency (i.e., messages are delivered through…
Many robotics applications benefit from being able to compute multiple locally optimal paths in a given configuration space. Examples include path planning for of tethered robots with cable-length constraints, systems involving cables,…
We investigate the dependence of the approximation capacity of deep residual networks on its depth in a continuous dynamical systems setting. This can be formulated as the general problem of quantifying the minimal time-horizon required to…
In this work, we study distance metric learning (DML) for high dimensional data. A typical approach for DML with high dimensional data is to perform the dimensionality reduction first before learning the distance metric. The main…
Robotic locomotion often relies on sequenced gaits to efficiently convert control input into desired motion. Despite extensive studies on gait optimization, achieving smooth and efficient gait transitions remains challenging. In this paper,…
Terrain geometry is, in general, non-smooth, non-linear, non-convex, and, if perceived through a robot-centric visual unit, appears partially occluded and noisy. This work presents the complete control pipeline capable of handling the…
We study the problem of bipedal robot navigation in complex environments with uncertain and rough terrain. In particular, we consider a scenario in which the robot is expected to reach a desired goal location by traversing an environment…
Geodesics deviation equation (GDE) is itroduced. In "adiabatic" approximation exact solution of the GDE if found. Perturbation theory in general case is formulated. Geometrical criterion of local instability which may lead to chaos is…
We show that both Teichmuller space (with the Teichmuller metric) and the mapping class group (with a word metric) have geodesic divergence that is intermediate between the linear rate of flat spaces and the exponential rate of hyperbolic…
We propose a new formulation and learning strategy for computing the Wasserstein geodesic between two probability distributions in high dimensions. By applying the method of Lagrange multipliers to the dynamic formulation of the optimal…
In this paper, the quantum corrections to the kinematics of geometry, specifically geodesics, are presented. This is done by employing the path integral over the geodesics. Interestingly, the geodesics do not see any modifications in this…
We introduce an algorithm which can be directly used to feasible and optimum search in linear programming. Starting from an initial point the algorithm iteratively moves a point in a direction to resolve the violated constraints. At the…
Dynamic subspace estimation, or subspace tracking, is a fundamental problem in statistical signal processing and machine learning. This paper considers a geodesic model for time-varying subspaces. The natural objective function for this…