Related papers: Toward a Functional Geometric Algebra for Natural …
Much of the success of deep learning is drawn from building architectures that properly respect underlying symmetry and structure in the data on which they operate - a set of considerations that have been united under the banner of…
Text embeddings have become central to computational social science and psychology, enabling scalable measurement of meaning and mixed-method inference. Yet most representation learning is optimized and evaluated for prediction and…
Clifford algebras have broad applications in science and engineering. The use of Clifford algebras can be further promoted in these fields by availability of computational tools that automate tedious routine calculations. We offer an…
Starting from the hypothesis that knowledge in semantic space is organized along structured manifolds, we argue that this geometric structure renders the space explorable. By traversing it and using the resulting continuous representations…
In the field of natural language understanding, the intersection of neural models and graph meaning representations (GMRs) remains a compelling area of research. Despite the growing interest, a critical gap persists in understanding the…
Retrieval-Augmented Generation (RAG) enhances Large Language Models (LLMs) by grounding responses in external knowledge during inference. However, conventiona RAG systems under-perform on structured tabular data, largely due to coarse…
Functional geometry is a framework using concepts from geometry to understand the invariance of amplitudes in quantum field theory under a large class of field redefinitions, including those involving derivatives. It is inspired by…
We extend the construction of [19] by introducing spaces of generalized tensor fields on smooth manifolds that possess optimal embedding and consistency properties with spaces of tensor distributions in the sense of L. Schwartz. We thereby…
Large Language Models (LLMs) have demonstrated substantial efficacy in advancing graph-structured data analysis. Prevailing LLM-based graph methods excel in adapting LLMs to text-rich graphs, wherein node attributes are text descriptions.…
This is a simple way rigorously to construct Grassmann, Clifford and Geometric Algebras, allowing degenerate bilinear forms, infinite dimension, using fields or certain modules (characteristic 2 with limitation) - and characterize the…
Semantic parsing is the task of producing a structured meaning representation for natural language utterances or questions. Recent research has pointed out that the commonly-used sequence-to-sequence (seq2seq) semantic parsers struggle to…
Clifford algebras are important structures in Geometric Algebra and Quantum Mechanics. They have allowed a formalization of the primitive operators in Quantum Theory. The algebras are built over vector spaces with dimension a power of 2…
The canonical quantisation of General Relativity including matter on a spacetime manifold in the globally hyperbolic setting involves in particular the representation theory of the spatial diffeomorphism group (SDG), and/or its Lie algebra…
Incorporating external knowledge bases in traditional retrieval-augmented generation (RAG) relies on parsing the document, followed by querying a language model with the parsed information via in-context learning. While effective for…
Graph Neural Networks (GNNs) have empowered the advance in graph-structured data analysis. Recently, the rise of Large Language Models (LLMs) like GPT-4 has heralded a new era in deep learning. However, their application to graph data poses…
Traditional power theories and one of their most important concepts --apparent power-- are still a source of debate and, as shown in the literature, they present several flaws that misinterpret the power-transfer phenomena under distorted…
A hard hadron-collider event is treated here as a single geometric object - the kinematics and the discrete object-type labels of all reconstructed final-state particles encoded in one multivector $\evMV\in\Cl(1,3)\otimes\Vflav$ - rather…
We present a geometric approach to defining an algebra $\hat{\mathcal G}(M)$ (the Colombeau algebra) of generalized functions on a smooth manifold $M$ containing the space ${\mathcal D}'(M)$ of distributions on $M$. Based on differential…
Convex analysis and Gaussian probability are tightly connected, as mostly evident in the theory of linear regression. Our work introduces an algebraic perspective on such relationship, in the form of a diagrammatic calculus of string…
The algebra of diffeomorphisms derived from general coordinate transformations on commuting coordinates is represented by differential operators on noncommutative spaces. The algebra remains unchanged, the comultiplication however is…