Related papers: Hologram Reasoning for Solving Algebra Problems wi…
We investigate the problem of multiplex graph embedding, that is, graphs in which nodes interact through multiple types of relations (dimensions). In recent years, several methods have been developed to address this problem. However, the…
Dynamic geometry systems (DGS) have become basic tools in many areas of geometry as, for example, in education. Geometry Automated Theorem Provers (GATP) are an active area of research and are considered as being basic tools in future…
The goal of this paper is to provide computational tools able to find a solution of a system of polynomial inequalities. The set of inequalities is reformulated as a system of polynomial equations. Three different methods, two of which…
Graph representations of solid state materials that encode only interatomic distance lack geometrical resolution, resulting in degenerate representations that may map distinct structures to equivalent graphs. Here we propose a hypergraph…
Higher-order graph neural networks (HOGNNs) and the related architectures from Topological Deep Learning are an important class of GNN models that harness polyadic relations between vertices beyond plain edges. They have been used to…
Geometric embedding methods have shown to be useful for multi-hop reasoning on knowledge graphs by mapping entities and logical operations to geometric regions and geometric transformations, respectively. Geometric embeddings provide direct…
Geometric algebra (GA) is a mathematical tool for geometric computing, providing a framework that allows a unified and compact approach to geometric relations which in other mathematical systems are typically described using different more…
An important problem in geometric reasoning is to find the configuration of a collection of geometric bodies so as to satisfy a set of given constraints. Recently, it has been suggested that this problem can be solved efficiently by…
Multi-hop question answering (MHQA) requires integrating knowledge scattered across multiple passages to derive the correct answer. Traditional retrieval-augmented generation (RAG) methods primarily focus on coarse-grained textual semantic…
We introduce a finite element method for numerical upscaling of second order elliptic equations with highly heterogeneous coefficients. The method is based on a mixed formulation of the problem and the concepts of the domain decomposition…
Graph similarity learning (GSL), also referred to as graph matching in many scenarios, is a fundamental problem in computer vision, pattern recognition, and graph learning. However, previous GSL methods assume that graphs are homogeneous…
Multigrid methods are well suited to large massively parallel computer architectures because they are mathematically optimal and display excellent parallelization properties. Since current architecture trends are favoring regular compute…
Motivated by the understanding of holography as realized in tensor networks, we develop a bulk procedure that can be interpreted as generating a sequence of coarse-grained holographic states. The coarse-graining procedure involves…
Large language models (LLMs) are being increasingly explored for graph tasks. Despite their remarkable success in text-based tasks, LLMs' capabilities in understanding explicit graph structures remain limited, particularly with large…
Homomorphism is a key mapping technique between graphs that preserves their structure. Given a graph and a pattern, the subgraph homomorphism problem involves finding a mapping from the pattern to the graph, ensuring that adjacent vertices…
We present a systematic, algebraically based, design methodology for efficient implementation of computer programs optimized over multiple levels of the processor/memory and network hierarchy. Using a common formalism to describe the…
A method for converting the geometrical problem of rectangle packing to an algebraic problem of solving a system of polynomial equations is described.
Large language models (LLMs) have demonstrated impressive reasoning abilities in complex tasks. However, they lack up-to-date knowledge and experience hallucinations during reasoning, which can lead to incorrect reasoning processes and…
The application of contemporary artificial intelligence techniques to address geometric problems and automated deductive proof has always been a grand challenge to the interdiscipline field of mathematics and artificial Intelligence. This…
The aim of this paper is to develop an algebraic multigrid method to solve eigenvalue problems based on the combination of the multilevel correction scheme and the algebraic multigrid method for linear equations. Our approach uses the…