The standard procedure for computing the persistent homology of a filtered simplicial complex is the matrix reduction algorithm. Its output is a particular decomposition of the total boundary matrix, from which the persistence diagrams and generating cycles can be derived. Persistence diagrams are known to vary continuously with respect to their input, which motivates the algorithmic study of persistence computations for time-varying filtered complexes. Computationally, simulating persistence dynamically can be reduced to maintaining a valid decomposition under adjacent transpositions in the filtration order. In practice, the quadratic scaling in the number of such transpositions often makes this maintenance procedure slower than simply computing the decomposition from scratch, limiting the application of dynamic persistence to relatively small data sets. In this work, we propose a coarser strategy for maintaining the decomposition over a 1-parameter family of filtrations. Our first result is an analysis of a simple linear-time strategy which reduces the number of column operations needed to simulate persistence across a fixed homotopy by at most a factor of 2. We show a modification of this technique which maintains only a sublinear number of valid states, as opposed to a quadratic number of states, and we provide tight lower bounds for this technique. Finally, we present results showing that the decrease in operations to compute diagrams across a family of filtrations is proportional to the difference between the expected quadratic number of states, and the proposed sublinear coarsening. Applications to multi-dimensional persistence and crocker stacks are also presented.

翻译：计算过滤单纯复形的持续同调的标准流程是矩阵约简算法。其输出是总边界矩阵的一种特定分解，由此可推导出持续图与生成圈。已知持续图随输入连续变化，这激发了对时变过滤复形持续计算的算法研究。从计算角度看，动态模拟持续同调可归结为在过滤序的相邻置换下维护有效分解。实际应用中，此类置换数量呈二次方增长，常使得维护过程比从头计算分解更耗时，限制了动态持续同调方法在较小数据集上的应用。本研究提出一种更粗粒度的策略，用于维护一族单参数过滤的分解。首先，我们分析一种线性时间策略的效能在固定同伦下模拟持续同调所需的列操作次数最多降低2倍。其次展示该技术的改进版本仅维护次线性数量的有效状态（而非二次方数量），并给出该技术的严格下界。最后，我们证明在过滤族中计算持续图的操作次数减少程度与预期的二次方状态数与所提出的次线性粗化策略之间的差值成正比。此外还给出了该技术在多维持续同调和克罗克堆栈中的应用。