论文标题
LG振动
The LG Fibration
论文作者
论文摘要
深度学习严重影响了整个研究和行业中数据的应用的应用，但是，它们缺乏严格的数学基础，这会创造出算法结果实际上无法逆转的情况。在本文中，我们通过$ s^{2^n-1} $和$ s^n $之间的拓扑连接，在$ \ mathbb {r}^{2^n} $和$ \ mathbb {r}^{n+1} $之间呈现几乎可逆的映射。在整篇文章中，我们利用多复合旋转组和多球坐标的代数来定义两个地图：第一个是从$ s^{2^n-1} $到$ \ $ \ displayStyle \ otimes^n_ {k = 1}的收缩。因此（2）$ to $ s^{n} $。这些形成一个复合图，我们称为LG纤维。类似于使用超复合几何形状从$ s^{（（2n-1）} \ mapsto cp^n $产生的HOPF纤维化，我们的纤维使用多复合几何形状来投影$ s^{2^n-1} $上的$ s^n $。我们还研究了LG振动的代数特性，最终得出了距离差函数，以确定哪些矢量对在转换下具有不变的内部产物。 LG振动在机器学习和AI上有应用，类似于HOPF纤维在自适应无人机控制中的当前应用。此外，几乎所有元素的LG振动的能力都允许开发机器学习算法，这些算法可能避免了当前遇到当代方法的不确定性和可重复性问题。本文的主要结果是一种新颖的方法，即从$ s^{2^n-1} $减少到$ s^n $，它可以扩展数学和AI的研究，包括但不限于Spheres，Algebraic拓扑，机器学习，机器学习和Algebraic cology。
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 $\mathbb{R}^{2^n}$ and $\mathbb{R}^{n+1}$ via a topological connection between $S^{2^n-1}$ and $S^n$. Throughout the paper we utilize the algebra of Multicomplex rotation groups and polyspherical coordinates to define two maps: the first is a contraction from $S^{2^n-1}$ to $\displaystyle \otimes^n_{k=1} SO(2)$, and the second is a projection from $\displaystyle \otimes^n_{k=1} SO(2)$ to $S^{n}$. Together these form a composite map that we call the LG Fibration. In analogy to the generation of Hopf Fibration using Hypercomplex geometry from $S^{(2n-1)} \mapsto CP^n$, our fibration uses Multicomplex geometry to project $S^{2^n-1}$ onto $S^n$. 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 $S^{2^n-1}$ to $S^n$, 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.
