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Network geometry, characterized by nodes with associated latent variables, is a fundamental feature of real-world networks. Still, when only the network edges are given, it may be difficult to assess whether the network contains an…
Neural networks are usually not the tool of choice for nonparametric high-dimensional problems where the number of input features is much larger than the number of observations. Though neural networks can approximate complex multivariate…
Large-scale eigenvalue problems arise in various fields of science and engineering and demand computationally efficient solutions. In this study, we investigate the subspace approximation for parametric linear eigenvalue problems, aiming to…
The Fermat-Weber location problem requires finding a point in $\mathbb{R}^n$ that minimizes the sum of weighted Euclidean distances to $m$ given points. An iterative solution method for this problem was first introduced by E. Weiszfeld in…
The aim of this article is to analyze numerical schemes using two-layer neural networks withinfinite width for the resolution of high-dimensional Schr{\"o}dinger eigenvalue problems with smoothinteraction potentials and Neumann boundary…
We revisit the minimum-link path problem: Given a polyhedral domain and two points in it, connect the points by a polygonal path with minimum number of edges. We consider settings where the vertices and/or the edges of the path are…
Pairwise comparison of graphs is key to many applications in Machine learning ranging from clustering, kernel-based classification/regression and more recently supervised graph prediction. Distances between graphs usually rely on…
One way to study the combinatorics of finite metric spaces is to study the betweenness relation associated with the metric space. In the hypergraph metrization problem, one has to find and characterize metric betweennesses whose collinear…
The paper is devoted to geometrical investigation of the Gromov-Hausdorff distance on the classes of all metric spaces and of all bounded metric spaces. The main attention is paid to pass connectivity questions. The pass connected…
One of the most intriguing findings in the structure of neural network landscape is the phenomenon of mode connectivity: For two typical global minima, there exists a path connecting them without barrier. This concept of mode connectivity…
Reducing dimension redundancy to find simplifying patterns in high-dimensional datasets and complex networks has become a major endeavor in many scientific fields. However, detecting the dimensionality of their latent space is challenging…
A network is said to have the properties of a small world if a suitably defined average distance between any two nodes is proportional to the logarithm of the number of nodes, $N$. In this paper, we present a novel derivation of the…
Networks with underlying metric spaces attract increasing research attention in network science, statistical physics, applied mathematics, computer science, sociology, and other fields. This attention is further amplified by the current…
The minimal network problem is a classical topic in geometric measure theory and the calculus of variations, which aims to find networks of minimal length connecting given points. Most classical results are established in the Euclidean…
The Gromov-Hausdorff distance ($d_\mathrm{GH}$) provides a natural way of quantifying the dissimilarity between two given metric spaces. It is known that computing $d_\mathrm{GH}$ between two finite metric spaces is NP-hard, even in the…
Developing efficient methods for solving parametric partial differential equations is crucial for addressing inverse problems. This work introduces a Least-Squares-based Neural Network (LS-Net) method for solving linear parametric PDEs. It…
Given a random sample of points from some unknown density, we propose a data-driven method for estimating density level sets under the r-convexity assumption. This shape condition generalizes the convexity property. However, the main…
This paper studies Minimum Spanning Trees under incomplete information for its vertices. We assume that no information is available on the precise placement of vertices so that it is only known that vertices belong to some neighborhoods…
In this paper, the proximal point algorithm for quasi-convex minimization problem in nonpositive curvature metric spaces is studied. We prove $\Delta$-convergence of the generated sequence to a critical point (which is defined in the text)…
Complex networks has been a hot topic of research over the past several years over crossing many disciplines, starting from mathematics and computer science and ending by the social and biological sciences. Random graphs were studied to…