标题： MCbiF：通过双参数持久同调测量多尺度聚类中的拓扑自相关性 显示英文标题

标题： MCbiF: Measuring Topological Autocorrelation in Multiscale Clusterings via 2-Parameter Persistent Homology

摘要： 数据集通常具有内在的多尺度结构，在不同粗细层次上都有有意义的描述。 这样的数据集自然可以描述为多分辨率聚类，即不一定是跨尺度的层次化分区序列。 为了分析和比较这些序列，我们使用拓扑数据分析工具，并定义了多尺度聚类双滤链（MCbiF），这是一个抽象单纯复形的二维参数滤链，它编码了跨尺度的聚类交集模式。 MCbiF可以解释为桑基图的高阶扩展，并在层次化序列中退化为树状图。 我们证明了MCbiF的多参数持久同调（MPH）产生了一个有限表示且块可分解的模，其稳定希尔伯特函数表征了分区序列的拓扑自相关性。 特别是，在零维时，MPH捕捉了分区细化顺序的违反情况，而在一维时，MPH捕捉了跨尺度的聚类之间的高阶不一致。 我们通过实验展示了MCbiF希尔伯特函数作为下游机器学习任务的拓扑特征图的使用。 MCbiF特征图在合成的非层次化分区序列上的回归和分类任务中优于基于信息的基线特征。 我们还展示了MCbiF在真实世界数据中的应用，以测量野生小鼠社会分组模式随时间变化的非层次性。 显示英文摘要

摘要： Datasets often possess an intrinsic multiscale structure with meaningful descriptions at different levels of coarseness. Such datasets are naturally described as multi-resolution clusterings, i.e., not necessarily hierarchical sequences of partitions across scales. To analyse and compare such sequences, we use tools from topological data analysis and define the Multiscale Clustering Bifiltration (MCbiF), a 2-parameter filtration of abstract simplicial complexes that encodes cluster intersection patterns across scales. The MCbiF can be interpreted as a higher-order extension of Sankey diagrams and reduces to a dendrogram for hierarchical sequences. We show that the multiparameter persistent homology (MPH) of the MCbiF yields a finitely presented and block decomposable module, and its stable Hilbert functions characterise the topological autocorrelation of the sequence of partitions. In particular, at dimension zero, the MPH captures violations of the refinement order of partitions, whereas at dimension one, the MPH captures higher-order inconsistencies between clusters across scales. We demonstrate through experiments the use of MCbiF Hilbert functions as topological feature maps for downstream machine learning tasks. MCbiF feature maps outperform information-based baseline features on both regression and classification tasks on synthetic sets of non-hierarchical sequences of partitions. We also show an application of MCbiF to real-world data to measure non-hierarchies in wild mice social grouping patterns across time.