Many topological data analysis (TDA) pipelines compute large collections of persistence diagrams, yet vectorizations and kernel methods discard the rank-induced implication relations among persistence intervals that are essential for faithful structural comparison and interpretability. We introduce higher-order persistence diagrams, a recursive construction in which containment relations among persistence intervals define higher-order persistence intervals. This construction performs comparison and aggregation directly on persistence diagrams and preserves interval-level structure. We use harmonic analysis to reduce frequency-space evaluations of aggregated diagrams to zeta transforms. This reduction avoids explicit construction of higher-order diagrams and replaces quadratic pair enumeration with nearly linear-time evaluation. Experiments on random network models show substantial speedups over explicit aggregation. Anonymized code is available at https://anonymous.4open.science/r/higher-order-persistence-8201.

翻译：许多拓扑数据分析（TDA）流程生成大量持续图，但向量化方法和核方法忽略了持续区间之间由秩诱导的蕴含关系，而这些关系对于忠实的结构比较和可解释性至关重要。我们引入高阶持续图，这是一种递归构造，其中持续区间之间的包含关系定义了高阶持续区间。该构造直接在持续图上进行比较和聚合，并保留了区间级结构。我们使用调和分析将聚合图的频域评估简化为zeta变换。这种简化避免了显式构造高阶图，并将二次对枚举替换为近线性时间评估。在随机网络模型上的实验表明，相较于显式聚合实现了显著加速。匿名代码可从 https://anonymous.4open.science/r/higher-order-persistence-8201 获取。