The LG Fibration

Deep Learning has significantly impacted the application of data-to-decision throughout research and industry, however, they lack a rigorous mathematical foundation, which creates situations where algorithmic results fail to be practically invertible. In this paper we present a nearly invertible mapping between ℝ2n and ℝn+1 via a topological connection between S2n-1 and Sn. Throughout the paper we utilize the algebra of Multicomplex rotation groups and polyspherical coordinates to define two maps: the first is a contraction from S2n-1 to displaystyle otimesnk=1 SO(2), and the second is a projection from displaystyle otimesnk=1 SO(2) to Sn. Together these form a composite map that we call the LG Fibration. In analogy to the generation of Hopf Fibration using Hypercomplex geometry from S(2n-1) mapsto CPn, our fibration uses Multicomplex geometry to project S2n-1 onto Sn. We also investigate the algebraic properties of the LG Fibration, ultimately deriving a distance difference function to determine which pairs of vectors have an invariant inner product under the transformation. The LG Fibration has applications to Machine Learning and AI, in analogy to the current applications of Hopf Fibrations in adaptive UAV control. Furthermore, the ability to invert the LG Fibration for nearly all elements allows for the development of Machine Learning algorithms that may avoid the issues of uncertainty and reproducibility that currently plague contemporary methods. The primary result of this paper is a novel method of nearly invertible geometric dimensional reduction from S2n-1 to Sn, which has the capability to extend the research in both mathematics and AI, including but not limited to the fields of homotopy groups of spheres, algebraic topology, machine learning, and algebraic biology.