Related papers: Random geometric graphs with smooth kernels: sharp…
A \emph{uniform random intersection graph} $G(n,m,k)$ is a random graph constructed as follows. Label each of $n$ nodes by a randomly chosen set of $k$ distinct colours taken from some finite set of possible colours of size $m$. Nodes are…
We find conditions for the connectivity of inhomogeneous random graphs with intermediate density. Our results generalize the classical result for G(n, p), when p = c log n/n. We draw n independent points X_i from a general distribution on a…
Consider a random geometric 2-dimensional simplicial complex $X$ sampled as follows: first, sample $n$ vectors $\boldsymbol{u_1},\ldots,\boldsymbol{u_n}$ uniformly at random on $\mathbb{S}^{d-1}$; then, for each triple $i,j,k \in [n]$, add…
We give upper and lower bounds on the information-theoretic threshold for community detection in the stochastic block model. Specifically, let $k$ be the number of groups, $d$ be the average degree, the probability of edges between vertices…
Causal discovery amounts to learning a directed acyclic graph (DAG) that encodes a causal model. This model selection problem can be challenging due to its large combinatorial search space, particularly when dealing with non-parametric…
This paper carries out a large dimensional analysis of a variation of kernel ridge regression that we call \emph{centered kernel ridge regression} (CKRR), also known in the literature as kernel ridge regression with offset. This modified…
The global clustering coefficient serves as a powerful metric for the structural analysis and comparison of complex networks. Random geometric graphs offer a realistic framework for representing the spatial constraints and geometry often…
Kernel methods are fundamental tools in machine learning that allow detection of non-linear dependencies between data without explicitly constructing feature vectors in high dimensional spaces. A major disadvantage of kernel methods is…
Edges of an image are considered a crucial type of information. These can be extracted by applying edge detectors with different methodology. Edge detection is a vital step in computer vision tasks, because it is an essential issue for…
The Grassmannian manifold G(k, n) serves as a fundamental tool in signal processing, computer vision, and machine learning, where problems often involve classifying, clustering, or comparing subspaces. In this work, we propose a…
We study fast algorithms for computing fundamental properties of a positive semidefinite kernel matrix $K \in \mathbb{R}^{n \times n}$ corresponding to $n$ points $x_1,\ldots,x_n \in \mathbb{R}^d$. In particular, we consider estimating the…
We study a generalization of the well-known model of broadcasting on trees. Consider a directed acyclic graph (DAG) with a unique source vertex $X$, and suppose all other vertices have indegree $d\geq 2$. Let the vertices at distance $k$…
Real-world networks, like social networks or the internet infrastructure, have structural properties such as large clustering coefficients that can best be described in terms of an underlying geometry. This is why the focus of the…
Rhombohedral graphite (RG) shows strong correlations in its topological flat band and is pivotal for exploring emergent, correlated electronic phenomena. One key advantage is the enhancement of electronic interactions with the increase in…
The threshold-$k$ metric dimension ($\mathrm{Tmd}_k$) of a graph is the minimum number of sensors -- a subset of the vertex set -- needed to uniquely identify any vertex in the graph, solely based on its distances from the sensors, when the…
We investigate some topological properties of random geometric complexes and random geometric graphs on Riemannian manifolds in the thermodynamic limit. In particular, for random geometric complexes we prove that the normalized counting…
Several emerging post-Bayesian methods target a probability distribution for which an entropy-regularised variational objective is minimised. This increased flexibility introduces a computational challenge, as one loses access to an…
We obtain upper bounds for the estimation error of Kernel Ridge Regression (KRR) for all non-negative regularization parameters, offering a geometric perspective on various phenomena in KRR. As applications: 1. We address the multiple…
We apply the discrepancy method and a chaining approach to give improved bounds on the coreset complexity of a wide class of kernel functions. Our results give randomized polynomial time algorithms to produce coresets of size…
The problems of detecting and recovering planted structures/subgraphs in Erd\H{o}s-R\'{e}nyi random graphs, have received significant attention over the past three decades, leading to many exciting results and mathematical techniques.…