The average distance from a node to all other nodes in a graph, or from a query point in a metric space to a set of points, is a fundamental quantity in data analysis. The inverse of the average distance, known as the (classic) closeness centrality of a node, is a popular importance measure in the study of social networks. We develop novel structural insights on the sparsifiability of the dista...

In this paper we study the average distance in weighted graphs. More precisely, we consider assignments of families of non-negative weights to the edges. The aim is to maximise (minimise, respectively) the average distance in the resulting weighted graph. Two variants of the problem are considered depending on whether the collection of weights is fixed or not. The main results of this paper are...

High Dimension Low Sample Size statistical analysis is becoming increasingly important in a wide range of applied contexts. In such situations, it is seen that the popular Support Vector Machine suffers from “data piling” at the margin, which can diminish generalizability. This leads naturally to the development of Distance Weighted Discrimination, which is based on Second Order Cone Programmin...

A new method is introduced for describing, visualizing, and inspecting data spaces. It is based on an adapted version of the Fast Exact Euclidean Distance (FEED) transform. It computes a description of the complete data space based on partial data. Combined with a metric, a true Weighted Distance Map (WDM) can be computed, which can define a probability space. Subsequently, distances between da...

We aim here at obtaining bounds on the first nonzero eigenvalue for selfadjoint boundary value problems on a weighted network by means of equilibrium measures, that include the study of Dirichlet, Neumann and Mixed problems. We also show the sharpness of these bounds throughout the analysis of some examples. In particular we emphasize the case of distance-regular graphs and we show that the obt...

High Dimension Low Sample Size statistical analysis is becoming increasingly important in a wide range of applied contexts. In such situations, it is seen that the appealing discrimination method called the Support Vector Machine can be improved. The revealing concept is “data piling” at the margin. This leads naturally to the development of “Distance Weighted Discrimination,” which also is bas...

Distance weighted discrimination (DWD) was originally proposed to handle the data piling issue in the support vector machine. In this paper, we consider the sparse penalized DWD for high-dimensional classification. The state-of-the-art algorithm for solving the standard DWD is based on second-order cone programming, however such an algorithm does not work well for the sparse penalized DWD with ...

We concern the problem of modifying the edge lengths of a tree in minimum total cost so that the prespecified p vertices become the p-maxian with respect to the new edge lengths. This problem is called the inverse p-maxian problem on trees. Gassner proposed efficient combinatorial alogrithm to solve the the inverse 1-maxian problem on trees in 2008. For the problem with p ≥ 2, we claim that the...

We concern the problem of modifying the edge lengths of a tree in minimum total cost so that the prespecified p vertices become the p-maxian with respect to the new edge lengths. This problem is called the inverse p-maxian problem on trees. Gassner proposed efficient combinatorial alogrithm to solve the the inverse 1-maxian problem on trees in 2008. For the problem with p ≥ 2, we claim that the...