Related papers: Intersection patterns of convex sets via simplicia…

The 1913 Helly's theorem states that any family ${\cal K}$ of $n\geq d+1$ convex sets in ${\mathbb R}^d$ can be pierced by a single point if and only if any $d+1$ of ${\cal K}$'s elements can. In 2002 Alon, Kalai, Matou\v{s}ek and Meshulam…

Complex systems are often driven by higher-order interactions among multiple units, naturally represented as hypergraphs. Understanding dependency structures within these hypergraphs is crucial for understanding and predicting the behavior…

This paper presents a novel approach for the differentiable rendering of convex polyhedra, addressing the limitations of recent methods that rely on implicit field supervision. Our technique introduces a strategy that combines…

While clusters of galaxies are considered one of the most important cosmological probes, the standard spherical modelling of the dark matter and the intracluster medium is only a rough approximation. Indeed, it is well established both…

Confounder selection is perhaps the most important step in the design of observational studies. A number of criteria, often with different objectives and approaches, have been proposed, and their validity and practical value have been…

We make a systematic study of the focal surface of a congruence of lines in the projective space. Using differential techniques together with techniques from intersection theory, we reobtain in particular all the invariants of the focal…

Closed form expressions are given for computing the parameters and vectors that identify and define the $n-1$ dimensional conic section that results from the intersection of a hyperplane with an $n$-dimensional conic section: cone,…

A central problem in topological data analysis is that of computing the homology of a given simplicial complex. Said complexes can have arbitrary large number of simplices, as can happen, for example, if the space is the Rips-Vietoris or…

In this paper we introduce a new model of random simplicial complexes depending on multiple probability parameters. This model includes the well-known Linial - Meshulam random simplicial complexes and random clique complexes as special…

Networks are often studied as graphs, where the vertices stand for entities in the world and the edges stand for connections between them. While relatively easy to study, graphs are often inadequate for modeling real-world situations,…

A new simple geometrical interpretation of complex numbers is presented. It differs from their usual interpretation as points in the complex plane. From the new point of view the complex numbers are rather operations on vectors than points.…

Complex systems consist of interacting units whose interactions may be pairwise, involving two units, or higher-order, involving more than two units simultaneously. Graphs capture pairwise interactions and represent such systems as…

Let X,Y be finite sets and T a set of functions from X -> Y which we will call "tableaux". We define a simplicial complex whose facets, all of the same dimension, correspond to these tableaux. Such "tableau complexes" have many nice…

A good classification method should yield more accurate results than simple heuristics. But there are classification problems, especially high-dimensional ones like the ones based on image/video data, for which simple heuristics can work…

This paper describes a framework for analysing matches in multiple data sets. The framework described is quite general and can be applied to a variety of problems where matches are to be found in data surveyed at a number of locations (or…

The differential cross-section for the reflection of light beams off rigid bodies obtained by the rotation of a generic derivable convex function is calculated. The calculation is developed using elementary notions of calculus and is…

In this paper, we propose a method for image-set classification based on convex cone models. Image set classification aims to classify a set of images, which were usually obtained from video frames or multi-view cameras, into a target…

The goal of this paper is to show that generalizing the notion of frequent patterns can be useful in extending association analysis to more complex higher order patterns. To that end, we describe a general framework for modeling a complex…

A new line of research on the lasso exploits the beautiful geometric fact that the lasso fit is the residual from projecting the response vector $y$ onto a certain convex polytope. This geometric picture also allows an exact geometric…

The matrix of canonical differential equations consists of the 1-$\mathrm{d}\log$-form coefficients obtained by projecting ($n$+1)-$\mathrm{d}\log$-forms onto $n$-$\mathrm{d}\log$-form master integrands. With dual form in relative…