$1$-parameter persistent homology, a cornerstone in Topological Data Analysis (TDA), studies the evolution of topological features such as connected components and cycles hidden in data. It has been applied to enhance the representation power of deep learning models, such as Graph Neural Networks (GNNs). To enrich the representations of topological features, here we propose to study $2$-parameter persistence modules induced by bi-filtration functions. In order to incorporate these representations into machine learning models, we introduce a novel vector representation called Generalized Rank Invariant Landscape \textsc{Gril} for $2$-parameter persistence modules. We show that this vector representation is $1$-Lipschitz stable and differentiable with respect to underlying filtration functions and can be easily integrated into machine learning models to augment encoding topological features. We present an algorithm to compute the vector representation efficiently. We also test our methods on synthetic and benchmark graph datasets, and compare the results with previous vector representations of $1$-parameter and $2$-parameter persistence modules.

翻译：1-参数持续同调作为拓扑数据分析（TDA）的基石，研究数据中隐藏的连通分量、环等拓扑特征的演化过程。该技术已被应用于增强图神经网络（GNNs）等深度学习模型的表征能力。为丰富拓扑特征的表示，本文提出研究由双过滤函数诱导的2-参数持续模。为将这些表示集成到机器学习模型中，我们引入一种新型向量表示——广义秩不变景观（GRIL），用于表征2-参数持续模。我们证明该向量表示在底层过滤函数下具有1-Lipschitz稳定性与可微性，可便捷地融入机器学习模型以增强拓扑特征编码。我们提出该向量表示的高效计算算法，并在合成数据集与基准图数据集上进行测试，将结果与1-参数及2-参数持续模的已有向量表示进行对比。