More Like this

1. Finding shortest and nearly shortest path nodes in large substantially incomplete networks by hyperbolic mapping

   https://doi.org/10.1038/s41467-022-35181-w  

   Kitsak, Maksim; Ganin, Alexander; Elmokashfi, Ahmed; Cui, Hongzhu; Eisenberg, Daniel A.; Alderson, David L.; Korkin, Dmitry; Linkov, Igor (December 2023, Nature Communications)

   Abstract Dynamic processes on networks, be it information transfer in the Internet, contagious spreading in a social network, or neural signaling, take place along shortest or nearly shortest paths. Computing shortest paths is a straightforward task when the network of interest is fully known, and there are a plethora of computational algorithms for this purpose. Unfortunately, our maps of most large networks are substantially incomplete due to either the highly dynamic nature of networks, or high cost of network measurements, or both, rendering traditional path finding methods inefficient. We find that shortest paths in large real networks, such as the network of protein-protein interactions and the Internet at the autonomous system level, are not random but are organized according to latent-geometric rules. If nodes of these networks are mapped to points in latent hyperbolic spaces, shortest paths in them align along geodesic curves connecting endpoint nodes. We find that this alignment is sufficiently strong to allow for the identification of shortest path nodes even in the case of substantially incomplete networks, where numbers of missing links exceed those of observable links. We demonstrate the utility of latent-geometric path finding in problems of cellular pathway reconstruction and communication security. 

   more » « less
2. Elastic Shape Analysis of Surfaces with Second-Order Sobolev Metrics: A Comprehensive Numerical Framework

   https://doi.org/10.1007/s11263-022-01743-0  

   Hartman, Emmanuel; Sukurdeep, Yashil; Klassen, Eric; Charon, Nicolas; Bauer, Martin (May 2023, International Journal of Computer Vision)

   Abstract This paper introduces a set of numerical methods for Riemannian shape analysis of 3D surfaces within the setting of invariant (elastic) second-order Sobolev metrics. More specifically, we address the computation of geodesics and geodesic distances between parametrized or unparametrized immersed surfaces represented as 3D meshes. Building on this, we develop tools for the statistical shape analysis of sets of surfaces, including methods for estimating Karcher means and performing tangent PCA on shape populations, and for computing parallel transport along paths of surfaces. Our proposed approach fundamentally relies on a relaxed variational formulation for the geodesic matching problem via the use of varifold fidelity terms, which enable us to enforce reparametrization independence when computing geodesics between unparametrized surfaces, while also yielding versatile algorithms that allow us to compare surfaces with varying sampling or mesh structures. Importantly, we demonstrate how our relaxed variational framework can be extended to tackle partially observed data. The different benefits of our numerical pipeline are illustrated over various examples, synthetic and real. 

   more » « less
3. Finding Your Way: Shortest Paths on Networks

   https://doi.org/10.3389/frym.2021.631045  

   Rexin, Teresa; Porter, Mason A. (December 2021, Frontiers for Young Minds)

   Traveling to different destinations is a major part of our lives. We visit a variety of locations both during our daily lives and when we are on vacation. How can we find the best way to navigate from one place to another? Perhaps we can test all of the different ways of traveling between two places, but another method is to use mathematics and computation to find a shortest path between them. In this article, we discuss how to construct shortest paths and introduce Dijkstra’s algorithm to minimize the total cost of a path, where the cost may be the travel distance, the travel time, or some other quantity. We also discuss how to use shortest paths in the real world to save time and increase traveling efficiency. 

   more » « less
4. Accelerated primal-dual methods for convex-strongly-concave saddle point problems

   Khalafi, Mohammad; Boob, Digvijay (July 2023, Proceedings of Machine Learning Research)

   We investigate a primal-dual (PD) method for the saddle point problem (SPP) that uses a linear approximation of the primal function instead of the standard proximal step, resulting in a linearized PD (LPD) method. For convex-strongly concave SPP, we observe that the LPD method has a suboptimal dependence on the Lipschitz constant of the primal function. To fix this issue, we combine features of Accelerated Gradient Descent with the LPD method resulting in a single-loop Accelerated Linearized Primal-Dual (ALPD) method. ALPD method achieves the optimal gradient complexity when the SPP has a semi-linear coupling function. We also present an inexact ALPD method for SPPs with a general nonlinear coupling function that maintains the optimal gradient evaluations of the primal parts and significantly improves the gradient evaluations of the coupling term compared to the ALPD method. We verify our findings with numerical experiments. 

   more » « less
5. Topo-Geometrically Distinct Path Computation Using Neighborhood-Augmented Graph, and Its Application to Path Planning for a Tethered Robot in 3-D

   Sahin, Alp; Bhattacharya, Subhrajit (November 2024, IEEE transactions on robotics)

   Many robotics applications benefit from being able to compute multiple geodesic paths in a given configuration space. Existing paradigm is to use topological path planning, which can compute optimal paths in distinct topological classes. However, these methods usually require nontrivial geometric constructions, which are prohibitively expensive in 3-D, and are unable to distinguish between distinct topologically equivalent geodesics that are created due to high-cost/curvature regions or prismatic obstacles in 3-D. In this article, we propose an approach to compute k geodesic paths using the concept of a novel neighborhood-augmented graph, on which graph search algorithms can compute multiple optimal paths that are topo-geometrically distinct. Our approach does not require complex geometric constructions, and the resulting paths are not restricted to distinct topological classes, making the algorithm suitable for problems where finding and distinguishing between geodesic paths are of interest. We demonstrate the application of our algorithm to planning shortest traversible paths for a tethered robot in 3-D with cable-length constraint. 

   more » « less