arXiv research

Infinite fractal tree solves shortest connection problem.

For a translation surface, we define the systole to be the length of the shortest saddle connection. We give a characterization of the maxima of the systole function on a stratum, and give a family of examples providing local but nonglobal maxima on each stratum of genus at least three. We further study the relation be…

Improved bounds on shortest geodesics with self-intersections on hyperbolic surfaces.

Study shows shortest geodesic length on certain manifolds is limited by volume, diameter, and cover elements.

In this paper, we show that for any closed 4-dimensional simply-connected Riemannian manifold $M$ with Ricci curvature $|Ric|\leq 3$, volume $vol(M)>v>0$, and diameter $diam(M)<D$, the length of a shortest closed geodesic is bounded by a function $F(v,D)$ which only depends on $v$ and $D$. The proofs of our result are …

Geodesics connect model modes in neural network loss landscapes.

Algorithms find second and third shortest non-trivial closed walks on surfaces.

Study examines how information flows in Indian stock market during crises.

The paper finds bounds on shortest dense curves on surfaces.

Study geodesics and shortest arcs on Lie groups with specific metrics.

Shortest geodesic on curved spheres is no longer than 3 times the diameter.

The paper studies the shortest closed multi-geodesics on hyperbolic surfaces as their genus grows.

Study geodesics and shortest arcs on Lie groups with specific metrics.

Generative Flow Networks solve shortest path problems in graphs.

Deep learning approximates shortest path distances in large graphs.

The problem of multiple surface clustering is a challenging task, particularly when the surfaces intersect. Available methods such as Isomap fail to capture the true shape of the surface nearby the intersection and result in incorrect clustering. The Isomap algorithm uses the shortest path between points. The main draw…

The paper finds shortest geodesic bounds on orbifolds with diameter limits.

Study on shortest arcs on hyperbolic surfaces with boundary.

We give a lower bound for the length of a non-trivial geodesic loop on a simply-connected and compact manifold of even dimension with a non-reversible Finsler metric of positive flag curvature. Harris and Paternain use this estimate in their recent paper [HP] to give a geometric characterization of dynamically convex F…

Study shortest non-simple geodesics on 2-orbifolds, finding unique shortest curve.

Study shortest non-separating curves on non-orientable surfaces, proving NP-hardness and tractability.

Shortest non-simple closed geodesics on hyperbolic surfaces found.

Graph embedding method captures both local and global network structure.

Sharp bounds found on shortest geodesic on punctured spheres.

The classical theorem of Fáry states that every planar graph can be represented by an embedding in which every edge is represented by a straight line segment. We consider generalizations of Fáry's theorem to surfaces equipped with Riemannian metrics. In this setting, we require that every edge is drawn as a shortest pa…

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.

Study shows shortest periodic geodesic on hyperbolic orbisphere complements figure-eight knot.

We compute the number of systoles, the shortest simple closed geodesics and 2-systoles, the second shortest simple closed geodesics on hyperbolic surfaces homeomorphic to once-punctured torus and four-punctured sphere.

Consider a weighted or unweighted k-nearest neighbor graph that has been built on n data points drawn randomly according to some density p on R^d. We study the convergence of the shortest path distance in such graphs as the sample size tends to infinity. We prove that for unweighted kNN graphs, this distance converges …

EntroPath learns manifold geometry from diffusion paths.

New data-driven Cartan connection tracks complex vascular structures.

We introduce efficient algorithms which achieve nearly optimal regrets for the problem of stochastic online shortest path routing with end-to-end feedback. The setting is a natural application of the combinatorial stochastic bandits problem, a special case of the linear stochastic bandits problem. We show how the diffi…

Explicit bounds found for shortest orthogeodesics and volumes of hyperbolic manifolds.

Any finite configuration of curves with minimal intersections on a surface is a configuration of shortest geodesics for some Riemannian metric on the surface. The metric can be chosen to make the lengths of these geodesics equal to the number of intersections along them.

Study shortest geodesics on flat cone spheres with conical singularities.

There have lately been several suggestions for parametrized distances on a graph that generalize the shortest path distance and the commute time or resistance distance. The need for developing such distances has risen from the observation that the above-mentioned common distances in many situations fail to take into ac…

Algorithm reduces regret in SSP problems with LFA.

Study bounds the length of shortest periodic geodesics on certain curved spaces.

In every conformal class of Finsler (or Riemannian) metrics on a closed manifold there exists a residual subset of Finsler metrics, such that, with respect to the residual Finsler metrics, in any non-trivial homotopy class of free loops there is precisely one shortest geodesic loop.

Study finds shortest geodesic loops on Stiefel manifold and calculates its injectivity radius.

The authors find geodesics, shortest arcs, diameter, cut locus, and conjugate sets for left-invariant sub-Riemannian metric on the Lie group SO(3), under condition that the metric is right-invariant relative to the Lie subgroup $SO(2)\subset SO(3)$.

New bounds on shortest geodesic loops on a sphere.

Landmark-based node embeddings approximate shortest path distances in random graphs.

There are many equivalent definitions of Riemannian geodesics. They are naturally generalised to sub-Riemannian manifold, but become non-equivalent. We give a review of different definitions of geodesics of a sub-Riemannian manifold and interrelation between them. We recall three variational definitions of geodesics as…

The study shows how many diameter directions in Besse manifolds relate to Blaschke manifolds.

In this paper, we prove some convergence results of a special case of optimistic policy iteration algorithm for stochastic shortest path problem. We consider both Monte Carlo and $TD(λ)$ methods for the policy evaluation step under the condition that the termination state will eventually be reached almost surely.

We present a simple, yet effective, approach to Semi-Supervised Learning. Our approach is based on estimating density-based distances (DBD) using a shortest path calculation on a graph. These Graph-DBD estimates can then be used in any distance-based supervised learning method, such as Nearest Neighbor methods and SVMs…