持续配对图：追踪选定同调基的变化
Persistence Pairing Graphs: Tracking Changes of Selected Homology Bases
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中文总结 AI 辅助
该研究提出依赖约简的持续配对图，用于记录选定同调类的变化，证明其性质并定义有界融合图差异，为持续同调提供了额外的结构描述。
中文摘要 AI 辅助
持续同调通过记录同调特征出现和消失的时刻来总结滤过，但生成的条形码无法确定首选同调基或首选闭链代表元。我们研究通过矩阵约简选定的基所携带的额外信息，并引入**持续配对图（persistence pairing graph）**，这是一种依赖于约简的有向图，用于记录当相关持续区间结束时，选定同调类如何重新表达。在固定一个滤过有限CW复形、一个系数域、一个与滤过兼容的胞腔序以及边界矩阵的约简分解后，该约简确定了选定的出生闭链。当相关持续区间结束时，其选定出生类的像在由区间存活至同一滤过值的选定类构成的基中具有唯一坐标，非零坐标定义了图的边。这些坐标也可通过与对偶上同调基的克罗内克配对得到。我们证明，在每个胞腔滤过阶段，活跃的选定出生闭链构成一个同调基；刻画了每个同调过渡的核与像；证明该图是有向无环图，每条边指向包含源区间的区间。在0维情况下，在标准约简约定下，该图与长者规则合并树一致。我们还证明，在持续对和条形码保持固定时，其边可以发生变化。最后，我们定义了有界融合图差异，并将普通持续稳定性控制的贡献与依赖于约简的图结构的贡献分离开来。因此，该图是相对于固定约简约定的描述符，而非单纯持续模的不变量。
英文摘要
Persistent homology summarizes a filtration by recording when homological features appear and disappear, but the resulting barcode does not determine a preferred homology basis or preferred cycle representatives. We study the additional information carried by a basis selected through matrix reduction and introduce the \emph{persistence pairing graph}, a reduction-dependent directed graph that records how selected homology classes are re-expressed when their associated persistence intervals end. After fixing a filtered finite CW complex, a coefficient field, a filtration-compatible cell order, and a reduced factorization of the boundary matrix, the reduction determines selected birth cycles. When an associated persistence interval ends, the image of its selected birth class has unique coordinates in the basis formed by selected classes whose intervals survive beyond the same filtration value, and the nonzero coordinates define the graph edges. These coordinates can also be recovered through the Kronecker pairing with a dual cohomology basis. We prove that the active selected birth cycles form a homology basis at every cellwise filtration stage, characterize the kernel and image of each homological transition, and show that the graph is directed and acyclic, with every edge pointing toward an interval containing the source interval. In dimension zero, under the standard reduction convention, the graph agrees with the elder rule merge tree. We also prove that its edges can change while the persistence pairs and barcode remain fixed. Finally, we define a bounded fused graph discrepancy and separate the contribution controlled by ordinary persistence stability from the reduction-dependent graph structure. The graph is therefore a descriptor relative to a fixed reduction convention, rather than an invariant of the persistence module alone.
