Exploring elliptic geometry in neural networks : The concept of elliptic neural networks (ENN)

Abstract

The exploration of non-Euclidean geometries in the context of neural network architectures presents a novel avenue for enhancing the processing of complex data structures. This paper introduces the concept of Elliptic Neural Networks (ENN), a framework that integrates the intrinsic properties of elliptic geometry into the fabric of neural computation. Characterized by constant positive curvature and closed geodesic paths, elliptic geometry provides a unique perspective for representing data, especially advantageous for configurations exhibiting cyclic and dense hierarchical interconnections. We begin by delineating the fundamental aspects of elliptic geometry, focusing on its impact on conventional notions of distance and global structural interpretation within data sets. This foundational understanding paves the way for our proposed mathematical schema for ENNs. Within this framework, we articulate the methodologies for embedding data in elliptic spaces and adapt neural network operations by encompassing neuron activation, signal propagation, and learning paradigms to conform to the topological idiosyncrasies of elliptic geometry. In this paper addressing the implementation of ENNs, we discuss the computational challenges and the development of novel algorithms requisite for navigating the elliptic geometric landscape. The prospective applications of ENNs are extensive and diverse, ranging from the enhancement of cyclic pattern recognition in time-series analysis to more sophisticated representations in graph-based data and natural language processing tasks. This theoretical exposition aims to set the groundwork for subsequent empirical research to validate the proposed model and assess its practical performance relative to conventional neural network architectures. By harnessing the distinctive characteristics of elliptic geometry, ENNs mark a significant stride towards enriching the arsenal of machine learning methodologies, potentially leading to more versatile and efficacious computational tools in data analysis and artificial intelligence.