Related papers: Geometric view of interval poset permutations
Numerous 6D pose estimation methods have been proposed that employ end-to-end regression to directly estimate the target pose parameters. Since the visible features of objects are implicitly influenced by their poses, the network allows…
We investigate the relationship between the geometry of token embeddings and their role in the next token prediction within transformer models. An important aspect of this connection uses the notion of empirical measure, which encodes the…
A grid polygon is a polygon whose vertices are points of a grid. We define an injective map between permutations of length n and a subset of grid polygons on n vertices, which we call consecutive-minima polygons. By the kernel method, we…
This article studies a large, general class of orthogonal polytopes which we may call "generic orthotopes". These objects emerged from a desire to represent a Coxeter complex by an orthogonal polytope that is particularly nice with respect…
This is both an expository and research paper where we advocate a systematic study of continuous analogues of finite partially ordered sets, convex polytopes, oriented matroids, arrangements of subspaces, finite simplicial complexes, and…
This expository article outlines the construction of De Concini-Procesi arrangement models and describes recent progress in understanding their significance from the algebraic, geometric, and combinatorial point of view. Throughout the…
The aim of this project is to attach a geometric structure to the ring of integers. It is generally assumed that the spectrum $\mathrm{Spec}(\mathbb{Z})$ defined by Grothendieck serves this purpose. However, it is still not clear what…
In this paper, we introduce the Grassmann tensor by tensor product of vectors and some basic terminology in tensor theory. Some basic properties of the Grassmann tensors are investigated and the tensor language is used to rewrite some…
We consider Lagrange interpolation on the set of finitely many intervals. This problem is closely related to the least deviating polynomial from zero on such sets. We will obtain lower and upper estimates for the corresponding Lebesgue…
The most recent trend in estimating the 6D pose of rigid objects has been to train deep networks to either directly regress the pose from the image or to predict the 2D locations of 3D keypoints, from which the pose can be obtained using a…
This note provides an overview of the notion of observable within the setting of multisymplectic geometry. We essentially follow the ideas described by F. H\'elein and J. Kouneiher [17] [18] [19] and in particular in keeping with the…
The dimension is a key measure of complexity of partially ordered sets. Small dimension allows succinct encoding. Indeed if $P$ has dimension $d$, then to know whether $x \leq y$ in $P$ it is enough to check whether $x\leq y$ in each of the…
Recently it has been shown that all non-trivial closed permutation groups containing the automorphism group of the random poset are generated by two types of permutations: the first type are permutations turning the order upside down, and…
In this paper we combine ideas from tolerance orders with recent work on OC interval orders. We consider representations of posets by unit intervals $I_v$ in which the interval endpoints ($L(v)$ and $R(v)$) may be open or closed as well as…
In this paper we present a simple framework to study various distance problems of permutations, including the transposition and block-interchange distance of permutations as well as the reversal distance of signed permutations. These…
For each composition $\vec{c}$ we show that the order complex of the poset of pointed set partitions $\Pi^{\bullet}_{\vec{c}}$ is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with…
We introduce a partial order structure on the set of interval orders of a given size, and prove that such a structure is in fact a lattice. We also provide a way to compute meet and join inside this lattice. Finally, we show that, if we…
Understanding the geometry and pose of objects in 2D images is a fundamental necessity for a wide range of real world applications. Driven by deep neural networks, recent methods have brought significant improvements to object pose…
We define interval spacing as the difference in the order statistics of data over a gap of some width. We derive its density, expected value, and variance for uniform, exponential, and logistic variates. We show that interval spacing is…
This paper considers the task of articulated human pose estimation of multiple people in real world images. We propose an approach that jointly solves the tasks of detection and pose estimation: it infers the number of persons in a scene,…