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1. On the estimation of latent distances using graph distances

   https://doi.org/10.1214/21-EJS1801  

   Arias-Castro, Ery; Channarond, Antoine; Pelletier, Bruno; Verzelen, Nicolas (January 2021, Electronic Journal of Statistics)

   null (Ed.)
2. Computing distances on Riemann surfaces

   https://doi.org/10.1088/1751-8121/ad653a  

   Stepanyants, Huck; Beardon, Alan; Paton, Jeremy; Krioukov, Dmitri (August 2024, Journal of Physics A: Mathematical and Theoretical)

   Abstract Riemann surfaces are among the simplest and most basic geometric objects. They appear as key players in many branches of physics, mathematics, and other sciences. Despite their widespread significance, how to compute distances between pairs of points on compact Riemann surfaces is surprisingly unknown, unless the surface is a sphere or a torus. This is because on higher-genus surfaces, the distance formula involves an infimum over infinitely many terms, so it cannot be evaluated in practice. Here we derive a computable distance formula for a broad class of Riemann surfaces. The formula reduces the infimum to a minimum over an explicit set consisting of finitely many terms. We also develop a distance computation algorithm, which cannot be expressed as a formula, but which is more computationally efficient on surfaces with high genuses. We illustrate both the formula and the algorithm in application to generalized Bolza surfaces, which are a particular class of highly symmetric compact Riemann surfaces of any genus greater than 1. 

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3. GLOBAL SENSITIVITY ANALYSIS MEASURES BASED ON STATISTICAL DISTANCES

   https://doi.org/10.1615/Int.J.UncertaintyQuantification.2021035424  

   Nandi, Souransu; Singh, Tarunraj (October 2021, International Journal for Uncertainty Quantification)

   null (Ed.)

   Global sensitivity analysis aims at quantifying and ranking the relative contribution of all the uncertain inputs of a mathematical model that impact the uncertainty in the output of that model, for any input-output mapping. Motivated by the limitations of the well-established Sobol' indices which are variance-based, there has been an interest in the development of non-moment-based global sensitivity metrics. This paper presents two complementary classes of metrics (one of which is a generalization of an already existing metric in the literature) which are based on the statistical distances between probability distributions rather than statistical moments. To alleviate the large computational cost associated with Monte Carlo sampling of the input-output model to estimate probability distributions, polynomial chaos surrogate models are proposed to be used. The surrogate models in conjunction with sparse quadrature-based rules, such as conjugate unscented transforms, permit efficient calculation of the proposed global sensitivity measures. Three benchmark sensitivity analysis examples are used to illustrate the proposed approach. 

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4. Characterizing Distances of Networks on the Tensor Manifold

   https://doi.org/10.1007/978-3-030-36687-2_79  

   Islam, Bipul; Liu, Ji; Sandhu, Romeil (November 2019, International Conference on Complex Networks and Their Applications)

   At the core of understanding dynamical systems is the ability to maintain and control the systems behavior that includes notions of robustness, heterogeneity, and/or regime-shift detection. Recently, to explore such functional properties, a convenient representation has been to model such dynamical systems as a weighted graph consisting of a finite, but very large number of interacting agents. This said, there exists very limited relevant statistical theory that is able cope with real-life data, i.e., how does perform analysis and/or statistics over a “family” of networks as opposed to a specific network or network-to-network variation. Here, we are interested in the analysis of network families whereby each network represents a “point” on an underlying statistical manifold. To do so, we explore the Riemannian structure of the tensor manifold developed by Pennec previously applied to Diffusion Tensor Imaging (DTI) towards the problem of network analysis. In particular, while this note focuses on Pennec definition of “geodesics” amongst a family of networks, we show how it lays the foundation for future work for developing measures of network robustness for regime-shift detection. We conclude with experiments highlighting the proposed distance on synthetic networks and an application towards biological (stem-cell) systems. 

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5. Sums of distances on graphs and embeddings into Euclidean space

   https://doi.org/10.1112/mtk.12198  

   Steinerberger, Stefan (July 2023, Mathematika)

   Abstract Let be a finite, connected graph. We consider a greedy selection of vertices: given a list of vertices , take to be any vertex maximizing the sum of distances to the vertices already chosen and iterate, keep adding the “most remote” vertex. The frequency with which the vertices of the graph appear in this sequence converges to a set of probability measures with nice properties. The support of these measures is, generically, given by a rather small number of vertices . We prove that this suggests that the graphGis, in a suitable sense, “m‐dimensional” by exhibiting an explicit 1‐Lipschitz embedding with good properties. 

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