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.

使用持久同源性来指导优化已成为拓扑数据分析的一种新颖应用。现有方法将持久性计算视为黑匣子，并且仅将梯度反向传播到特定对中涉及的单纯形上。我们展示了如何使用持久性计算中使用的循环和链来规定域的更大子集的梯度​​。特别是，我们表明，在作为一般损失的构建块的特殊情况下，该问题可以在线性时间内精确解决。这依赖于本文的另一个贡献，它消除了检查具有相同值的单纯形的阶乘数排列的需要。我们提出的实证实验表明了我们的算法的实际好处：优化所需的步骤数减少了一个数量级。