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。在需要针对离散数据开发新型数学工具的数据科学快速发展的背景下，构建连接矩阵的高效计算框架尤为重要。为此，经典连接矩阵理论已被适配至便于计算的组合框架中。本文提出一种高效的持久化类算法，用于从单纯复形上的给定组合（多）向量场计算连接矩阵。该算法仅需单次遍历，相较于需在每一层级执行两次遍历的隐式递归算法，显著改进了计算效率。整体而言，新算法比现有最优算法更简洁、直接且高效。由于该算法与持久化算法的相似性，可充分利用拓扑数据分析中的各类软件优化策略。