One of the most interesting tools that have recently entered the data science toolbox is topological data analysis (TDA). With the explosion of available data sizes and dimensions, identifying and extracting the underlying structure of a given dataset is a fundamental challenge in data science, and TDA provides a methodology for analyzing the shape of a dataset using tools and prospects from algebraic topology. However, the computational complexity makes it quickly infeasible to process large datasets, especially those with high dimensions. Here, we introduce a preprocessing strategy called the Characteristic Lattice Algorithm (CLA), which allows users to reduce the size of a given dataset as desired while maintaining geometric and topological features in order to make the computation of TDA feasible or to shorten its computation time. In addition, we derive a stability theorem and an upper bound of the barcode errors for CLA based on the bottleneck distance.

翻译：近年来进入数据科学工具库中最有趣的工具之一是拓扑数据分析（TDA）。随着可用数据规模和维度的爆炸式增长，识别并提取给定数据集的潜在结构成为数据科学中的基本挑战，而TDA提供了一种利用代数拓扑的工具和前景来分析数据集形状的方法。然而，计算复杂性使得处理大型数据集（尤其是高维数据集）迅速变得不可行。本文提出了一种名为特征格算法（CLA）的预处理策略，允许用户按需缩减给定数据集的大小，同时保留几何与拓扑特征，从而使TDA的计算变得可行或缩短其计算时间。此外，我们基于瓶颈距离推导了CLA的稳定性定理以及条形码误差的上界。