环面坐标：用格点归约解相关圆坐标
Toroidal Coordinates: Decorrelating Circular Coordinates With Lattice Reduction
摘要：
de Silva、Morozov和Vejdemo-Johansson的圆坐标算法将数据集与表示数据中的$1$维孔的上同调类一起作为输入;输出是从数据到捕获该孔的圆的映射，并且在适当的意义上具有最小能量。然而，当应用到几个上同调类，输出的圆值映射可以是“几何相关”，即使选择的上同调类是线性无关的。它表明，在原来的工作，相关性较低的地图，可以得到适当的整数线性组合的上同调类，与线性组合被选中的检查。在本文中，我们确定了一个正式的概念，圈值映射之间的几何相关，在黎曼流形的情况下，对应于狄利克雷形式，一个双线性形式来自狄利克雷能量。我们描述了一个系统的程序，从一组线性无关的上同调类的数据，低能量环面值映射。我们展示我们的程序与计算的例子。我们的主要算法基于计算数论中的Lenstra-Lenstra-Lovász算法。
英文摘要：
The circular coordinates algorithm of de Silva, Morozov, and Vejdemo-Johansson takes as input a dataset together with a cohomology class representing a $1$-dimensional hole in the data; the output is a map from the data into the circle that captures this hole, and that is of minimum energy in a suitable sense. However, when applied to several cohomology classes, the output circle-valued maps can be "geometrically correlated" even if the chosen cohomology classes are linearly independent. It is shown in the original work that less correlated maps can be obtained with suitable integer linear combinations of the cohomology classes, with the linear combinations being chosen by inspection. In this paper, we identify a formal notion of geometric correlation between circle-valued maps which, in the Riemannian manifold case, corresponds to the Dirichlet form, a bilinear form derived from the Dirichlet energy. We describe a systematic procedure for constructing low energy torus-valued maps on data, starting from a set of linearly independent cohomology classes. We showcase our procedure with computational examples. Our main algorithm is based on the Lenstra--Lenstra--Lovász algorithm from computational number theory.
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