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--Lov\'asz algorithm from computational number theory.

翻译：de Silva、Morozov和Vejdemo-Johansson提出的圆形坐标算法，以数据集及表示数据中一维空洞的上同调类作为输入，输出一个将该空洞捕获至圆环的映射，且该映射在适当意义下能量最小。然而，当应用于多个上同调类时，即使所选上同调类线性无关，输出的圆值映射仍可能呈现"几何相关"。原始工作表明，通过适当整系数线性组合上同调类（通过观察选取组合）可获得相关性更低的映射。本文给出圆值映射间几何相关性的形式化定义，在黎曼流形情形下，该定义对应由狄利克雷能量导出的双线性型——狄利克雷形式。我们描述了一种系统化方法，从一组线性无关的上同调类出发，在数据上构造低能量环面值映射，并通过计算实例展示该方法。主要算法基于计算数论中的Lenstra-Lenstra-Lovász算法。