分层地图的梯度抽样算法及其在拓扑数据分析中的应用
A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis
摘要：
我们引入了一种新的梯度下降算法，扩展了著名的梯度采样方法的分层光滑的目标函数，这被定义为局部Lipschitz函数，是光滑的一些定期片-所谓的层-周围的欧几里德空间的类。对于这类函数，我们的算法实现了次线性收敛速度。然后，我们将我们的方法应用于目标函数的基础上（扩展）持久的同源映射计算低星过滤器，这是一个核心工具的拓扑数据分析。为此，我们提出了一个有效的探索相应的分层使用的Cayley图的置换群。最后，我们提供了基准和新的拓扑优化问题，以证明我们的框架的实用性和适用性。
英文摘要：
We introduce a novel gradient descent algorithm extending the well-known Gradient Sampling methodology to the class of stratifiably smooth objective functions, which are defined as locally Lipschitz functions that are smooth on some regular pieces-called the strata-of the ambient Euclidean space. For this class of functions, our algorithm achieves a sub-linear convergence rate. We then apply our method to objective functions based on the (extended) persistent homology map computed over lower-star filters, which is a central tool of Topological Data Analysis. For this, we propose an efficient exploration of the corresponding stratification by using the Cayley graph of the permutation group. Finally, we provide benchmark and novel topological optimization problems, in order to demonstrate the utility and applicability of our framework.
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