论文标题
拓扑优化大步
Topological Optimization with Big Steps
论文作者
论文摘要
使用持续的同源性指导优化已成为拓扑数据分析的新应用。现有方法将持久性计算视为黑匣子，而仅在特定对涉及的简单上进行反向流动梯度。我们展示了如何使用持久计算中使用的循环和链来规定域较大子集的梯度。特别是，我们表明，在一种特殊情况下，它是一般损失的基础，可以在线性时间中精确解决问题。这取决于本文的另一项贡献，这消除了需要检查具有相同值的简单排列数量的需求。我们提出了表明我们算法的实际好处的经验实验：优化所需的步骤数量通过数量级减少。
Using persistent homology to guide optimization has emerged as a novel application of topological data analysis. Existing methods treat persistence calculation as a black box and backpropagate gradients only onto the simplices involved in particular pairs. We show how the cycles and chains used in the persistence calculation can be used to prescribe gradients to larger subsets of the domain. In particular, we show that in a special case, which serves as a building block for general losses, the problem can be solved exactly in linear time. This relies on another contribution of this paper, which eliminates the need to examine a factorial number of permutations of simplices with the same value. We present empirical experiments that show the practical benefits of our algorithm: the number of steps required for the optimization is reduced by an order of magnitude.
