发表机构
HSE University; Noeon Research（高等经济大学; 诺恩研究）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出 QF-树数据结构，用于高效存储旗复形除以子复形所得的商 CW 复形，并证明其尺寸线性、可局部应用，优于锥模型。
AI 中文摘要
Vietoris-Rips 滤流（在拓扑数据分析中标准）是旗复形，而单纯形树无需任何附加数据即可存储这些复形。本文探讨当旗复形 $K$ 被子复形 $A$ 除时，这种经济性保留了多少，其中 $A$ 的每个连通分量被压缩为一点。这样的商是一个 CW 复形，其胞腔是 $K\setminus A$ 的单纯形，但它们的附着映射不再是隐式的。我们证明，对于旗复形 $K$，当且仅当 $A$ 是旗复形时，商的面的序是严格分级的，并且存活的标记胞腔由维数至多 3 的胞腔决定。对于 $m$-旗对，阈值为 $2m+1$，当 $K$ 是旗复形时降至 $m+2$。仅当 $A$ 在 $K$ 中是满的时，指定的胞腔构成正则 CW 分解。这些结果证明了 QF-树的合理性：一个胞腔表，对于每个存活的单纯形，存储其有序的 $d+1$ 个面（其中折叠的面被标记），并通过商顶点词的 trie 索引。对于有界维数，其大小与存活单纯形数加上保留的出处信息成线性关系，我们推导并验证了一个简单公式，用于计算折叠比例，超过该比例时它比同伦等价的锥模型更小。由于折叠仅改变 $A$ 的闭星上的附着数据，QF-树也可在单纯形树内部局部应用。对于采样 Vietoris-Rips 区域中的球状 $A$，闭星是薄壳，且在每个采样半径下，中位紧凑预算低于锥模型。一个以 Gudhi 为模型的伴随库实现了 QF-树、其局部变体、具有局部商更新的可编辑层、粘合、圆盘附着、诱导映射、杯积、基本群表示和之字形持久性，并提供的实验将维护商的成本与在其上计算的代数成本分开。
英文摘要
Vietoris-Rips filtrations, which are standard in topological data analysis, are flag complexes, and a simplex tree stores these without any attaching data. In this paper we ask what survives of this economy when a flag complex $K$ is divided by a subcomplex $A$, each connected component of $A$ being crushed to a point. Such a quotient is a CW complex whose cells are the simplices of $K\setminus A$, but their attaching maps are no longer implicit. We show that for flag $K$ the face order of the quotient is strictly graded exactly when $A$ is flag, and that the surviving labelled cells are determined by those of dimension at most 3. For $m$-flag pairs the threshold is $2m+1$, and it drops to $m+2$ when $K$ is flag. The prescribed cells form a regular CW decomposition only when $A$ is full in $K$. These results justify the QF-tree: a cell table that stores, for each surviving simplex, its ordered list of $d+1$ facets with collapsed facets flagged, indexed by a trie of quotient-vertex words. For bounded dimension its size is linear in the number of surviving simplices plus the retained provenance, and we derive and verify a simple formula for the collapsed fraction above which it is smaller than the homotopy-equivalent cone model. Because a collapse changes the attaching data only on the closed star of $A$, the QF-tree can also be applied locally inside a simplex tree. For a ball-shaped $A$ in the sampled Vietoris--Rips regime the closed star is a thin shell, and the median compact budget is below the cone model at every sampled radius. An accompanying library, modelled on Gudhi, implements the QF-tree, its local variant, an editable layer with local quotient updates, gluing, disc attachment, induced maps, cup products, fundamental-group presentations and zigzag persistence, and provided experiments separate the cost of maintaining a quotient from the cost of the algebra computed on it.
