Connection matrices are a generalization of Morse boundary operators from the classical Morse theory for gradient vector fields. Developing an efficient computational framework for connection matrices is particularly important in the context of a rapidly growing data science that requires new mathematical tools for discrete data. Toward this goal, the classical theory for connection matrices has been adapted to combinatorial frameworks that facilitate computation. We develop an efficient persistence-like algorithm to compute a connection matrix from a given combinatorial (multi) vector field on a simplicial complex. This algorithm requires a single-pass, improving upon a known algorithm that runs an implicit recursion executing two-passes at each level. Overall, the new algorithm is more simple, direct, and efficient than the state-of-the-art. Because of the algorithm's similarity to the persistence algorithm, one may take advantage of various software optimizations from topological data analysis.

翻译：连接矩阵是经典莫尔斯理论中梯度向量场的莫尔斯边界算子的推广。在需要离散数据新数学工具的快速发展的数据科学背景下，开发连接矩阵的高效计算框架尤为重要。为实现这一目标，经典的连接矩阵理论已被改编为便于计算的组合框架。我们开发了一种类似保持性算法的高效算法，用于从单纯复形上的给定组合（多）向量场计算连接矩阵。该算法采用单次遍历，改进了已知的隐式递归算法（该算法在每一层级执行两次遍历）。总体而言，新算法比现有最优方法更简单、直接且高效。由于该算法与保持性算法的相似性，可以充分利用拓扑数据分析中的各种软件优化技术。