Related papers: Measuring Depth of Matroids
New sharp affine isoperimetric inequalities for volume decomposition functionals $X_{2}$ and $X_{3}$ in $\mathbb{R}^n$ are established. To fulfil this task, we prove the recursion formulas for volume decomposition functionals and find out…
The model theory based notion of the first order convergence unifies the notions of the left-convergence for dense structures and the Benjamini-Schramm convergence for sparse structures. It is known that every first order convergent…
In this paper, we first give a new result characterizing the strongly connected digraphs with a diameter equal to that of their line digraphs. Then, we introduce the concepts of the inner diameter and inner radius of a digraph and study…
Recent advances have markedly improved the cross-scene generalization of relative depth estimation, yet its practical applicability remains limited by the absence of metric scale, local inconsistencies, and low computational efficiency. To…
Understanding the relationship between the depth of a neural network and its representational capacity is a central problem in deep learning theory. In this work, we develop a geometric framework to analyze the expressivity of ReLU networks…
In this paper, the concept of the metric matrix is introduced to establish a concise and unified formulation for the inner product in barycentric coordinates. Building on this framework, we explore the intrinsic algebraic identities of…
Assessing the similarity of matrices is valuable for analyzing the extent to which data sets exhibit common features in tasks such as data clustering, dimensionality reduction, pattern recognition, group comparison, and graph analysis.…
This paper studies structural aspects of lattice path matroids, a class of transversal matroids that is closed under taking minors and duals. Among the basic topics treated are direct sums, duals, minors, circuits, and connected flats. One…
This work presents Prior Depth Anything, a framework that combines incomplete but precise metric information in depth measurement with relative but complete geometric structures in depth prediction, generating accurate, dense, and detailed…
Omnidirectional depth estimation presents a significant challenge due to the inherent distortions in panoramic images. Despite notable advancements, the impact of projection methods remains underexplored. We introduce Multi-Cylindrical…
In this paper, we propose a novel penetration metric, called deformable penetration depth PDd, to define a measure of inter-penetration between two linearly deforming tetrahedra using the object norm. First of all, we show that a distance…
We present a formal analysis, in Isabelle/HOL, of optimisation algorithms for matroids, which are useful generalisations of combinatorial structures that occur in optimisation, and greedoids, which are a generalisation of matroids. Although…
In this work, we present a panoramic metric depth foundation model that generalizes across diverse scene distances. We explore a data-in-the-loop paradigm from the view of both data construction and framework design. We collect a…
We aim to conduct a systematic mapping in the area of testing ML programs. We identify, analyze and classify the existing literature to provide an overview of the area. We followed well-established guidelines of systematic mapping to…
The recent increase of interest in the graph invariant called tree-depth and in its applications in algorithms and logic on graphs led to a natural question: is there an analogously useful "depth" notion also for dense graphs (say; one…
Matroid varieties are the closures in the Grassmannian of sets of points defined by specifying which Pl\"ucker coordinates vanish and which don't --- the set of nonvanishing Pl\"ucker coordinates forms a well-studied object called a…
We propose a framework for descriptively analyzing sets of partial orders based on the concept of depth functions. Despite intensive studies of depth functions in linear and metric spaces, there is very little discussion on depth functions…
In this paper, we propose a metric on the space of finite sets of trajectories for assessing multi-target tracking algorithms in a mathematically sound way. The main use of the metric is to compare estimates of trajectories from different…
In recent years, the emergence of foundation models for depth prediction has led to remarkable progress, particularly in zero-shot monocular depth estimation. These models generate impressive depth predictions; however, their outputs are…
In this paper, we define the notion of rigidity for linear electrical multiports and for matroid pairs. We show the parallel between the two and study the consequences of this parallel. We present applications to testing, using purely…