To analyze the topological properties of the given discrete data, one needs to consider a continuous transform called filtration. Persistent homology serves as a tool to track changes of homology in the filtration. The outcome of the topological analysis of data varies depending on the choice of filtration, making the selection of filtration crucial. Filtration learning is an attempt to find an optimal filtration that minimizes the loss function. Exact Multi-parameter Persistent Homology (EMPH) has been recently proposed, particularly for topological time-series analysis, that utilizes the exact formula of rank invariant instead of calculating it. In this paper, we propose a framework for filtration learning of EMPH. We formulate an optimization problem and propose an algorithm for solving the problem. We then apply the proposed algorithm to several classification problems. Particularly, we derive the exact formula of the gradient of the loss function with respect to the filtration parameters, which makes it possible to directly update the filtration without using automatic differentiation, significantly enhancing the learning process.

翻译：为分析给定离散数据的拓扑性质，需引入称为滤过的连续变换。持久同调作为追踪滤过中同调变化的工具，其分析结果因滤过的选择而异，这使得滤过的选取至关重要。滤过学习旨在寻找使损失函数最小化的最优滤过。精确多参数持久同调（EMPH）近期被提出，尤其适用于拓扑时间序列分析，该方法直接利用秩不变量的精确公式而非通过计算获得。本文提出EMPH的滤过学习框架，构建优化问题并设计求解算法。我们将所提算法应用于若干分类问题，特别推导了损失函数关于滤过参数梯度的精确表达式，从而无需借助自动微分即可直接更新滤过参数，显著提升了学习效率。