We study the algorithmic complexity of computing persistent homology of a randomly generated filtration. Specifically, we prove upper bounds for the average fill-in (number of non-zero entries) of the boundary matrix on \v{C}ech, Vietoris--Rips and Erd\H{o}s--R\'enyi filtrations after matrix reduction. Our bounds show that the reduced matrix is expected to be significantly sparser than what the general worst-case predicts. Our method is based on previous results on the expected Betti numbers of the corresponding complexes. We establish a link between these results and the fill-in of the boundary matrix. In the $1$-dimensional case, our bound for \v{C}ech and Vietoris--Rips complexes is asymptotically tight up to a logarithmic factor. We also provide an Erd\H{o}s--R\'enyi filtration realising the worst-case.

翻译：我们研究了随机生成滤过的持久同调计算的算法复杂度。具体而言，我们证明了在Čech、Vietoris–Rips和Erdős–Rényi滤过中，经过矩阵约简后边界矩阵的平均填充（非零条目数）的上界。我们的结果表明，约简后的矩阵在预期上显著稀疏，远优于一般最坏情况的预测。该方法基于对应复形的预期贝蒂数的已有结果，我们将这些结果与边界矩阵的填充联系起来。在一维情况下，对于Čech和Vietoris–Rips复形，我们的上界在渐近意义下紧（仅相差一个对数因子）。我们还提供了一个实现最坏情况的Erdős–Rényi滤过。