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.

本文介绍了一种用于拓扑图形分析中的持续图的优化方法，在保持稳定性的同时可以生成更健壮的极大值，并且可以将持续图中的关键和接近关键的单形分布可视化。

全局拓扑功能优化的快速稳健方法

A Fast and Robust Method for Global Topological Functional Optimization

We develop algorithms for sampling from a probability distribution on a
submanifold embedded in Rn. Applications are given to the evaluation of
algorithms in 'Topological Statistics'; to goodness of fit tests in exponential
families and to Neyman's smooth test. This article is partially expository,
giving an introduction to the tools of geometric measure theory.

本文提出了从 Rn 中嵌入的子流形上的概率分布中进行抽样的算法，并应用于 “拓扑统计” 中算法的评估，指数族的拟合度检验以及 Neyman 的平滑检验。该文章部分是阐述性的，介绍了几何测度论的工具。

从流形中进行采样

Sampling From A Manifold

We describe a new methodology for studying persistence of topological
features across a family of spaces or point-cloud data sets, called zigzag
persistence. Building on classical results about quiver representations, zigzag
persistence generalises the highly successful theory of persistent homology and
addresses several situations which are not covered by that theory. In this
paper we develop theoretical and algorithmic foundations with a view towards
applications in topological statistics.

介绍了一种研究空间或点云数据集中拓扑特征持久性的新方法，称为 zigzag persistence，该方法建立在关于 quiver representions 的经典结果之上，推广了持久性同调理论并解决了该理论未涉及的一些问题。本文发展了这种理论方法和算法基础，旨在应用于拓扑统计学。