What is the "right" topological invariant of a large point cloud X? Prior research has focused on estimating the full persistence diagram of X, a quantity that is very expensive to compute, unstable to outliers, and far from a sufficient statistic. We therefore propose that the correct invariant is not the persistence diagram of X, but rather the collection of persistence diagrams of many small subsets. This invariant, which we call "distributed persistence," is trivially parallelizable, more stable to outliers, and has a rich inverse theory. The map from the space of point clouds (with the quasi-isometry metric) to the space of distributed persistence invariants (with the Hausdorff-Bottleneck distance) is a global quasi-isometry. This is a much stronger property than simply being injective, as it implies that the inverse of a small neighborhood is a small neighborhood, and is to our knowledge the only result of its kind in the TDA literature. Moreover, the quasi-isometry bounds depend on the size of the subsets taken, so that as the size of these subsets goes from small to large, the invariant interpolates between a purely geometric one and a topological one. Lastly, we note that our inverse results do not actually require considering all subsets of a fixed size (an enormous collection), but a relatively small collection satisfying certain covering properties that arise with high probability when randomly sampling subsets. These theoretical results are complemented by two synthetic experiments demonstrating the use of distributed persistence in practice.
翻译:X 大点云的“ 右” 表层变量是什么? 先前的研究侧重于估算 X 的完整持久性图, 这个数量对于计算非常昂贵,对于外星来说不稳定,而且远没有足够统计数据。 因此, 我们建议正确的变量不是 X 的持久性图, 而是收集许多小子集的持久性图。 这个我们称之为“ 分布性持久性” 的变量是微不足道的平行的, 更稳定于外星体, 并且具有丰富的反向理论。 从点云空间( 与准测量度测量度相比) 到分布性惯性惯性( 与Hausdorf- Bottleneck 距离相比) 空间的地图是一个全球的准测量性。 这比仅仅具有暗示作用的小区区还要强得多, 因为它意味着我们所了解的只有这种类型的结果。 此外, 从点云层云的空间( 与准测量度测量度测量度测量量相比), 其准测量范围取决于子体的大小, 因此, 其精确度的精确度范围从一个层次到一个相对的精确的层次, 需要从一个小的层次中, 从一个小的层次中, 从一个小的层次中, 从一个小的层次的层次中, 从一个小的层次的层次到一个小数, 从一个小的层次, 从一个小的层次, 从一个我们从一个小到一个小的层次的层次, 从一个小到一个层次的层次, 从一个层次, 从一个小到一个层次, 从一个层次, 从一个开始, 从一个小到一个层次。