Given a persistence diagram with $n$ points, we give an algorithm that produces a sequence of $n$ persistence diagrams converging in bottleneck distance to the input diagram, the $i$th of which has $i$ distinct (weighted) points and is a $2$-approximation to the closest persistence diagram with that many distinct points. For each approximation, we precompute the optimal matching between the $i$th and the $(i+1)$st. Perhaps surprisingly, the entire sequence of diagrams as well as the sequence of matchings can be represented in $O(n)$ space. The main approach is to use a variation of the greedy permutation of the persistence diagram to give good Hausdorff approximations and assign weights to these subsets. We give a new algorithm to efficiently compute this permutation, despite the high implicit dimension of points in a persistence diagram due to the effect of the diagonal. The sketches are also structured to permit fast (linear time) approximations to the Hausdorff distance between diagrams -- a lower bound on the bottleneck distance. For approximating the bottleneck distance, sketches can also be used to compute a linear-size neighborhood graph directly, obviating the need for geometric data structures used in state-of-the-art methods for bottleneck computation.
翻译:根据一个含有美元点数的持久性图,我们给出了一个算法,产生一个以美元为单位的坚持性图序列,在瓶颈距离内与输入图相融合,其中美元为美元(加权)点数,这是对最接近的持久性图的2美元比方,与许多不同点相匹配。对于每个近似,我们预先计算美元和美元(i+1)美元之间的最佳匹配。也许令人惊讶的是,整个图表序列和匹配序列都可以以美元(n)空间表示。主要的方法是使用耐久性图的贪婪变换来给Hausdorff(加权)点数,并给这些子分配权重。我们给出了一个新的算法,以便有效地进行这种调整,尽管由于对角的影响,在持久性图中的点数隐含了很高的尺寸。草图结构也允许对Hausdorf之间的距离进行快速(线内时间)的近比值。主要方法是使用耐久性调整图的宽度调整,对于瓶度图的距离而言,也是用于直径的直径的直径,用于直径的平方平方位计算。