Related papers: Phase transition in noisy high-dimensional random …
We study the effect of the choice of embedding geometry on the entropy of random geometric graph ensembles with soft connection functions. First we show that when the connection range is small, the entropy is dependent only on the dimension…
Turing patterns, arising from the interplay between competing species of diffusive particles, has long been an important concept for describing non-equilibrium self-organization in nature, and has been extensively investigated in many…
We consider the random geometric graph on $n$ vertices drawn uniformly from a $d$--dimensional sphere. We focus on the sparse regime, when the expected degree is constant independent of $d$ and $n$. We show that, when $d$ is larger than $n$…
Estimating the state of an environment from high-dimensional, multimodal, and noisy observations is a fundamental challenge in reinforcement learning (RL). Traditional approaches rely on probabilistic models to account for the uncertainty,…
In this paper, we study the acquaintance time $\AC(G)$ defined for a connected graph $G$. We focus on $\G(n,r,p)$, a random subgraph of a random geometric graph in which $n$ vertices are chosen uniformly at random and independently from…
In this work we introduce Dynamic Random Geometric Graphs as a basic rough model for mobile wireless sensor networks, where communication distances are set to the known threshold for connectivity of static random geometric graphs. We…
A common observation in data-driven applications is that high-dimensional data have a low intrinsic dimension, at least locally. In this work, we consider the problem of point estimation for manifold-valued data. Namely, given a finite set…
In this paper, we consider the problem of testing the equality of two multivariate distributions based on geometric graphs constructed using the interpoint distances between the observations. These include the tests based on the minimum…
We analyze the asymptotic number and typical structure of claw-free graphs at constant edge densities. The first of our main results is a formula for the asymptotics of the logarithm of the number of claw-free graphs of edge density $\gamma…
We study the complexity of training neural network models with one hidden nonlinear activation layer and an output weighted sum layer. We analyze Gradient Descent applied to learning a bounded target function on $n$ real-valued inputs. We…
The problem of network-constrained averaging is to compute the average of a set of values distributed throughout a graph G using an algorithm that can pass messages only along graph edges. We study this problem in the noisy setting, in…
Autonomous (noise-agnostic) generative models, such as Equilibrium Matching and blind diffusion, challenge the standard paradigm by learning a single, time-invariant vector field that operates without explicit noise-level conditioning.…
Geometric phases, accumulated when a quantum system traces a cycle in quantum state space, do not depend on the parametrization of the cyclic path, but do depend on the path itself. In the presence of noise that deforms the path, the phase…
We introduce a new class of countably infinite random geometric graphs, whose vertices are points in a metric space, and vertices are adjacent independently with probability p if the metric distance between the vertices is below a given…
Consider a graph on $n$ uniform random points in the unit square, each pair being connected by an edge with probability $p$ if the inter-point distance is at most $r$. We show that as $n\to\infty$ the probability of full connectivity is…
The object of study is a soft random geometric graph with vertices given by a Poisson point process on a line and edges between vertices present with probability that has a polynomial decay in the distance between them. Various aspects of…
High-dimensional data are ubiquitous, with examples ranging from natural images to scientific datasets, and often reside near low-dimensional manifolds. Leveraging this geometric structure is vital for downstream tasks, including signal…
We study the graph alignment problem over two independent Erd\H{o}s-R\'enyi graphs on $n$ vertices, with edge density $p$ falling into two regimes separated by the critical window around $p_c=\sqrt{\log n/n}$. Our result reveals an…
The fault tolerance of random graphs with unbounded degrees with respect to connectivity is investigated, which relates to the reliability of wireless sensor networks with unreliable relay nodes. The model evaluates the network breakdown…
We consider random geometric graphs on the plane characterized by a non-uniform density of vertices. In particular, we introduce a graph model where $n$ vertices are independently distributed in the unit disc with positions, in polar…