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We introduce the intersection cohomology module of a matroid and prove that it satisfies Poincar\'e duality, the hard Lefschetz theorem, and the Hodge-Riemann relations. As applications, we obtain proofs of Dowling and Wilson's Top-Heavy…

Different types of two- and three-dimensional representations of a finite metric space are studied that focus on the accurate representation of the linear order among the distances rather than their actual values. Lower and upper bounds for…

The concept of representing a polytope that is associated with some combinatorial optimization problem as a linear projection of a higher-dimensional polyhedron has recently received increasing attention. In this paper (written for the…

This paper puts forward a new generalized polynomial dimensional decomposition (PDD), referred to as GPDD, comprising hierarchically ordered measure-consistent multivariate orthogonal polynomials in dependent random variables. Unlike the…

We derive a general upper bound for the number of incidences with $k$-dimensional varieties in ${\mathbb R}^d$. The leading term of this new bound generalizes previous bounds for the special cases of $k=1, k=d-1,$ and $k= d/2$, to every…

Modern applications of algebraic topology to point cloud data analysis have motivated active investigation of combinatorial clique complexes -- high-dimensional extensions of combinatorial graphs. We show that meaningful invariants of such…

We present a joint 3D pose and focal length estimation approach for object categories in the wild. In contrast to previous methods that predict 3D poses independently of the focal length or assume a constant focal length, we explicitly…

The paper introduces a generalization for known probabilistic models such as log-linear and graphical models, called here multiplicative models. These models, that express probabilities via product of parameters are shown to capture…

In this letter, we introduce several algorithms for geometry inpainting of 3D point clouds with large holes. The algorithms are examplar-based: hole filling is performed iteratively using templates near the hole boundary to find the best…

Iterated planar contact manifolds are a generalization of three dimensional planar contact manifolds to higher dimensions. We study some basic topological properties of iterated planar contact manifolds and discuss several examples and…

Deep Neural Networks achieve state-of-the-art results in many different problem settings by exploiting vast amounts of training data. However, collecting, storing and - in the case of supervised learning - labelling the data is expensive…

As a crucial task of autonomous driving, 3D object detection has made great progress in recent years. However, monocular 3D object detection remains a challenging problem due to the unsatisfactory performance in depth estimation. Most…

The aim of this survey is to outline the state of the art in research on a class of linearized polynomials with coefficients over finite fields, known as scattered polynomials. These have been studied in several contexts, such as in [A.…

Phylogenetic models have polynomial parametrization maps. For symmetric group-based models, Matsen studied the polynomial inequalities that characterize the joint probabilities in the image of these parametrizations. We employ this…

The geometry of rotations in dimensions 3, 4, and 5 is discussed using the matrix exponential map. Explicit closed formulas for the exponential of an antisymmetric matrix, as well as the logarithm of a rotation, are given for these…

Several incidence theorems of planar projective geometry are considered. It is demonstrated that generalizations of Pascal theorem due to M\"obius give rise to double cross-ratio equation and Hietarinta equation. The construction…

Spatial perception aims to estimate camera motion and scene structure from visual observations, a problem traditionally addressed through geometric modeling and physical consistency constraints. Recent learning-based methods have…

Recently introduced composition operator for credal sets is an analogy of such operators in probability, possibility, evidence and valuation-based systems theories. It was designed to construct multidimensional models (in the framework of…

Starting from the (apparently) elementary problem of deciding how many different topological spaces can be obtained by gluing together in pairs the faces of an octahedron, we will describe the central role played by hyperbolic geometry…

This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The…