Related papers: Geodesic Distance Estimation with Spherelets
Geodesic distance matrices can reveal shape properties that are largely invariant to non-rigid deformations, and thus are often used to analyze and represent 3-D shapes. However, these matrices grow quadratically with the number of points.…
Computing an array of all pairs of geodesic distances between the pixels of an image is time consuming. In the sequel, we introduce new methods exploiting the redundancy of geodesic propagations and compare them to an existing one. We show…
By "geodesic" we mean any sequence of vertices $(v_1,v_2,...,v_k)$ of a graph $G$ that constitute a shortest path from $v_1$ to $v_k$. We propose a novel, natural algorithm to enumerate all geodesics of $G$, and pit it (using Mathematica)…
Geodesic distance, sometimes called shortest path length, has proven useful in a great variety of applications, such as information retrieval on networks including treelike networked models. Here, our goal is to analytically determine the…
We give an algorithm for determining the distance between two vertices of the complex of curves. While there already exist such algorithms, for example by Leasure, Shackleton, and Webb, our approach is new, simple, and more effective for…
Machine learning problems have an intrinsic geometric structure as central objects including a neural network's weight space and the loss function associated with a particular task can be viewed as encoding the intrinsic geometry of a given…
We introduce a novel representation and optimization framework for discrete geodesics on triangle meshes that reduces artifacts of linear methods on uneven and coarse discretizations. Our method computes squared geodesic distances from…
Random geometric graphs are random graph models defined on metric measure spaces. A random geometric graph is generated by first sampling points from a metric space and then connecting each pair of sampled points independently with a…
Riemannian optimization uses local methods to solve optimization problems whose constraint set is a smooth manifold. A linear step along some descent direction usually leaves the constraints, and hence retraction maps are used to…
This paper describes continuous-space methodologies to estimate the collision probability, Euclidean distance and gradient between an ellipsoidal robot model and an environment surface modeled as a set of Gaussian distributions.…
The approximation of both geodesic distances and shortest paths on point cloud sampled from an embedded submanifold $\mathcal{M}$ of Euclidean space has been a long-standing challenge in computational geometry. Given a sampling resolution…
In algorithms for finite metric spaces, it is common to assume that the distance between two points can be computed in constant time, and complexity bounds are expressed only in terms of the number of points of the metric space. We…
We propose a novel method for determining the radius of a spherical surface based on the distances measured between points on this surface. We consider the most general case of determining the radius when the distances are measured with…
New geometric and computational analyses of power-weighted shortest-path distances (PWSPDs) are presented. By illuminating the way these metrics balance density and geometry in the underlying data, we clarify their key parameters and…
An $\epsilon$-approximate incidence between a point and some geometric object (line, circle, plane, sphere) occurs when the point and the object lie at distance at most $\epsilon$ from each other. Given a set of points and a set of objects,…
In this paper, we study differentially private (DP) algorithms for computing the geometric median (GM) of a dataset: Given $n$ points, $x_1,\dots,x_n$ in $\mathbb{R}^d$, the goal is to find a point $\theta$ that minimizes the sum of the…
We prove existence and uniqueness of the minimizer for the average geodesic distance to the points of a geodesically convex set on the sphere. This implies a corresponding existence and uniqueness result for an optimal algorithm for…
In this paper, we study the problem of computing Euclidean geodesic centers of a polygonal domain $\mathcal{P}$ with a total of $n$ vertices. We discover many interesting observations. We give a necessary condition for a point being a…
Distance metrics and their nonlinear variant play a crucial role in machine learning based real-world problem solving. We demonstrated how Euclidean and cosine distance measures differ not only theoretically but also in real-world medical…
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…