There has been recent interest in novel Clifford geometric invariants of
linear transformations. This motivates the investigation of such invariants for
a certain type of geometric transformation of interest in the context of root
systems, reflection groups, Lie groups and Lie algebras: the Coxeter
transformations. We perform exhaustive calculations of all Coxeter
transformations for $A_8$, $D_8$ and $E_8$ for a choice of basis of simple
roots and compute their invariants, using high-performance computing. This
computational algebra paradigm generates a dataset that can then be mined using
techniques from data science such as supervised and unsupervised machine
learning. In this paper we focus on neural network classification and principal
component analysis. Since the output -- the invariants -- is fully determined
by the choice of simple roots and the permutation order of the corresponding
reflections in the Coxeter element, we expect huge degeneracy in the mapping.
This provides the perfect setup for machine learning, and indeed we see that
the datasets can be machine learned to very high accuracy. This paper is a
pump-priming study in experimental mathematics using Clifford algebras, showing
that such Clifford algebraic datasets are amenable to machine learning, and
shedding light on relationships between these novel and other well-known
geometric invariants and also giving rise to analytic results.

这篇论文通过高性能计算，对 Coxeter 变换进行了全面计算，通过监督和无监督机器学习的技术，展示了 Clifford 代数数据集可以用于机器学习，并揭示了这些新颖几何不变量与其他众所周知的几何不变量之间的关系。

ADE Coxeter 元素的机器学习 Clifford 不变量

Machine Learning Clifford invariants of ADE Coxeter elements

In recent decades, a growing number of discoveries in fields of mathematics
have been assisted by computer algorithms, primarily for exploring large
parameter spaces that humans would take too long to investigate. As computers
and algorithms become more powerful, an intriguing possibility arises - the
interplay between human intuition and computer algorithms can lead to
discoveries of novel mathematical concepts that would otherwise remain elusive.
To realize this perspective, we have developed a massively parallel computer
algorithm that discovers an unprecedented number of continued fraction formulas
for fundamental mathematical constants. The sheer number of formulas discovered
by the algorithm unveils a novel mathematical structure that we call the
conservative matrix field. Such matrix fields (1) unify thousands of existing
formulas, (2) generate infinitely many new formulas, and most importantly, (3)
lead to unexpected relations between different mathematical constants,
including multiple integer values of the Riemann zeta function. Conservative
matrix fields also enable new mathematical proofs of irrationality. In
particular, we can use them to generalize the celebrated proof by Ap\'ery for
the irrationality of $\zeta(3)$. Utilizing thousands of personal computers
worldwide, our computer-supported research strategy demonstrates the power of
experimental mathematics, highlighting the prospects of large-scale
computational approaches to tackle longstanding open problems and discover
unexpected connections across diverse fields of science.

通过计算机算法发现大量连分数公式，并揭示了保守矩阵场这一新颖数学结构，统一了现有的数学公式，生成了无穷多个新公式，并引发了不同数学常数之间的意想不到的联系，同时可以用于证明无理性定理。实验证明了大规模计算方法在攻克长期悬而未决的问题并发现不同科学领域之间的意外联系中的潜力。