Related papers: Descriptive Proximities I: Properties and interpla…
Supervised machine learning involves approximating an unknown functional relationship from a limited dataset of features and corresponding labels. The classical approach to feature-based machine learning typically relies on applying linear…
The motivation for using qualitative shape descriptions is as follows: qualitative shape descriptions can implicitly act as a schema for measuring the similarity of shapes, which has the potential to be cognitively adequate. Then, shapes…
Overlap, also known as positivity, is a key condition for causal treatment effect estimation. Many popular estimators suffer from high variance and become brittle when features differ strongly across treatment groups. This is especially…
Rough set theory, a mathematical tool to deal with inexact or uncertain knowledge in information systems, has originally described the indiscernibility of elements by equivalence relations. Covering rough sets are a natural extension of…
Uniformity and proximity are two different ways for defining small scale structures on a set. Coarse structures are large scale counterparts of uniform structures. In this paper, motivated by the definition of proximity, we develop the…
Density-functional theory is a formally exact description of a many-body quantum system in terms of its density; in practice, however, approximations to the universal density functional are required. In this work, a model based on deep…
The Desargues property is well known in the context of projective geometry. An analogous property is presented in the context of both classical and Quantum Physics. In a classical context, the Desargues property implies that two logical…
Piecewise-linear (PL) Morse theory and discrete Morse theory are used in shape analysis tasks to investigate the topological features of discretized spaces. In spite of their common origin in smooth Morse theory, various notions of critical…
Decomposition spaces are a class of function spaces constructed out of well-behaved coverings and partitions of unity of a set. The structure of the covering of the set determines the properties of the decomposition space. Besov spaces,…
In this work, we study the correlation between attribute sets and the occurrence of dense subgraphs in large attributed graphs, a task we call structural correlation pattern mining. A structural correlation pattern is a dense subgraph…
We investigate the large intersection properties of the set of points that are approximated at a certain rate by a family of affine subspaces. We then apply our results to various sets arising in the metric theory of Diophantine…
The decomposition into interaction subspaces is a hierarchical decomposition of the spaces of cylindrical functions of a finite product space, also called factor spaces. It is an important construction in graphical models and a standard way…
This work presents a quantitative framework for describing the overcompleteness of a large class of frames. It introduces notions of localization and approximation between two frames $\mathcal{F} = \{f_i\}_{i \in I}$ and $\mathcal{E} =…
The paper considers a new quantitative-qualitative proximity measure for the features of information objects, where data enters a common information resource from several sources independently. The goal is to determine the possibility of…
In previous work, we introduced a method for modeling a configuration of objects in 2D and 3D images using a mathematical "medial/skeletal linking structure." In this paper, we show how these structures allow us to capture positional…
Proximity maps and regions are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in…
Interactions play a key role in understanding objects and scenes, for both virtual and real world agents. We introduce a new general representation for proximal interactions among physical objects that is agnostic to the type of objects or…
In a human-robot collaborative task where a robot helps its partner by finding described objects, the depth dimension plays a critical role in successful task completion. Existing studies have mostly focused on comprehending the object…
The visual world around us can be described as a structured set of objects and their associated relations. An image of a room may be conjured given only the description of the underlying objects and their associated relations. While there…
Detecting and exploiting similarities between seemingly distant objects is without doubt an important human ability. This paper develops \textit{from the ground up} an abstract algebraic and qualitative notion of similarity based on the…