Related papers: Orthologics for Cones
We study the model theoretic strength of various lattices that occur naturally in topology, like closed (semi-linear or semi-algebraic or convex) sets. The method is based on weak monadic second order logic and sharpens previous results by…
Ontologies are useful for automatic machine processing of domain knowledge as they represent it in a structured format. Yet, constructing ontologies requires substantial manual effort. To automate part of this process, large language models…
A closure system with the anti-exchange axiom is called a convex geometry. One geometry is called a sub-geometry of the other if its closed sets form a sublattice in the lattice of closed sets of the other. We prove that convex geometries…
We develop an algorithmic theory of convex optimization over discrete sets. Using a combination of algebraic and geometric tools we are able to provide polynomial time algorithms for solving broad classes of convex combinatorial…
In ontology-mediated querying, description logic (DL) ontologies are used to enrich incomplete data with domain knowledge which results in more complete answers to queries. However, the evaluation of ontology-mediated queries (OMQs) over…
In domains with high knowledge distribution a natural objective is to create principle foundations for collaborative interactive learning environments. We present a first mathematical characterization of a collaborative learning group, a…
We involve a certain propositional logic based on ortholattices. We characterize the implicational reduct of such a logic and we show that its algebraic counterpart is the so-called orthosemilattice. Properties of congruences and congruence…
Optimization is at the core of control theory and appears in several areas of this field, such as optimal control, distributed control, system identification, robust control, state estimation, model predictive control and dynamic…
Query embedding (QE) -- which aims to embed entities and first-order logical (FOL) queries in low-dimensional spaces -- has shown great power in multi-hop reasoning over knowledge graphs. Recently, embedding entities and queries with…
This dissertation explores applications of discrete geometry in mathematical neuroscience. We begin with convex neural codes, which model the activity of hippocampal place cells and other neurons with convex receptive fields. In Chapter 4,…
The following article treats about convex geometries which are lower semi-modular and join semi-distributive lattices. Firstly, it is shown that there is a class $K$ of infinite convex geometries which can be build out of finite ones by…
Geometric relational embeddings map relational data as geometric objects that combine vector information suitable for machine learning and structured/relational information for structured/relational reasoning, typically in low dimensions.…
A description of the environment cognition process by intelligent systems with a fixed set of system goals is suggested. Such a system is represented by the set of its goals only without any models of the system elements or the environment.…
Knowledge representation learning (KRL) aims to represent entities and relations in knowledge graph in low-dimensional semantic space, which have been widely used in massive knowledge-driven tasks. In this article, we introduce the reader…
The current learning systems typically lack the level of metacognitive awareness, self-directed learning, and time management skills. Most of the ontologically based learning management systems are in the proposed phase and those which are…
We propose an abstract definition of convex spaces as sets where one can take convex combinations in a consistent way. A priori, a convex space is an algebra over a finitary version of the Giry monad. We identify the corresponding Lawvere…
Ontologies comprising of concepts, their attributes, and relationships are used in many knowledge based AI systems. While there have been efforts towards populating domain specific ontologies, we examine the role of document structure in…
In applied mathematics and related disciplines, the modeling-simulation-optimization workflow is a prominent scheme, with mathematical models and numerical algorithms playing a crucial role. For these types of mathematical research data,…
The integration of optimization problems within neural network architectures represents a fundamental shift from traditional approaches to handling constraints in deep learning. While it is long known that neural networks can incorporate…
We study nonconvex optimization landscapes for learning overcomplete representations, including learning (i) sparsely used overcomplete dictionaries and (ii) convolutional dictionaries, where these unsupervised learning problems find many…