Topological statistics, in the form of persistence diagrams, are a class of shape descriptors that capture global structural information in data. The mapping from data structures to persistence diagrams is almost everywhere differentiable, allowing for topological gradients to be backpropagated to ordinary gradients. However, as a method for optimizing a topological functional, this backpropagation method is expensive, unstable, and produces very fragile optima. Our contribution is to introduce a novel backpropagation scheme that is significantly faster, more stable, and produces more robust optima. Moreover, this scheme can also be used to produce a stable visualization of dots in a persistence diagram as a distribution over critical, and near-critical, simplices in the data structure.
翻译:地形统计,以持久性图表的形式,是一组形状描述器,在数据中捕捉全球结构信息。从数据结构到持久性图表的映射几乎无处不在,让表层梯度反射到普通梯度。然而,作为一种优化地形功能的方法,这种回映方法成本昂贵、不稳定,并产生非常脆弱的opima。我们的贡献是引入一种新型的回映计划,其速度要快得多、更稳定,并产生更强大的Opima。此外,这个计划还可以用于生成一个稳定可视化的持久性图中点,作为数据结构中关键和接近临界的微粒的分布。