Related papers: Concentration phenomena in high dimensional geomet…
The stability of random variables can be generalized in any convex cone. In this case the principal results about the LePage representation and the domains of attraction are analogous but different to those well known for general Banach…
We study intersections of projective convex sets in the sense of Steinitz. In a projective space, an intersection of a nonempty family of convex sets splits into multiple connected components each of which is a convex set. Hence, such an…
Current work on human-machine alignment aims at understanding machine-learned latent spaces and their correspondence to human representations. G{\"a}rdenfors' conceptual spaces is a prominent framework for understanding human…
Enumeration of various types of lattice polygons and in particular polyominoes is of primary importance in many machine learning, pattern recognition, and geometric analysis problems. In this work, we develop a large deviation principle for…
In this paper we study the existence and uniqueness of best proximity points of cyclic contractions as well as the convergence of iterates to such proximity points. We do it from two different approaches, leading each one of them to…
In this paper we address the question whether in a given Banach space, a Chebyshev center of a nonempty bounded subset can be a farthest point of the set. Our exploration reveals that the answer depends on the convexity properties of the…
We consider convex sets whose modulus of convexity is uniformly quadratic. First, we observe several interesting relations between different positions of such ``2-convex'' bodies; in particular, the isotropic position is a finite…
Curvature properties of convex objects, such as strong convexity, are important in designing and analyzing convex optimization algorithms in the Hilbertian or Riemannian settings. In the case of the Hilbertian setting, strongly convex sets…
We study geometric properties of a random Gaussian short-time correlated velocity field by considering statistics of a passively advected metric tensor. That describes universal properties of fluctuations of tensor objects frozen into the…
Random graph models are used to describe the complex structure of real-world networks in diverse fields of knowledge. Studying their behavior and fitting properties are still critical challenges, that in general, require model specific…
This paper is devoted to a systematic study and characterizations of the fundamental notions of variational and strong variational convexity for lower semicontinuous functions. While these notions have been quite recently introduced by…
Recent developments in the physics of ultracold gases provide wide possibilities for reducing the dimensionality of space for magnetically or optically trapped atoms. The goal of these lectures is to show that regimes of quantum degeneracy…
Empirical and theoretical works show that the input weights of two-layer neural networks, when initialized with small values, converge towards isolated orientations. This phenomenon, referred to as condensation, indicates that the gradient…
High-dimensional representations for words, text, images, knowledge graphs and other structured data are commonly used in different paradigms of machine learning and data mining. These representations have different degrees of…
We propose novel randomized optimization methods for high-dimensional convex problems based on restrictions of variables to random subspaces. We consider oblivious and data-adaptive subspaces and study their approximation properties via…
We derive concentration inequalities for the spectral measure of large random matrices, allowing for certain forms of dependence. Our main focus is on empirical covariance (Wishart) matrices, but general symmetric random matrices are also…
In this paper, we formulate a simple algorithm that detects contours around a region of interest in an image. After an initial smoothing, the method is based on viewing an image as a topographic surface and finding convex and/or concave…
Network models with latent geometry have been used successfully in many applications in network science and other disciplines, yet it is usually impossible to tell if a given real network is geometric, meaning if it is a typical element in…
This paper deals with the problem of detecting non-isotropic high-dimensional geometric structure in random graphs. Namely, we study a model of a random geometric graph in which vertices correspond to points generated randomly and…
We investigate quantitative implications of the notion of log-concavity through a probabilistic interpretation. In particular, we derive concentration inequalities, moment and entropy bounds for random variables satisfying a precise degree…