$1$-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 (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. Further, we augment GNNs with GRIL features and observe an increase in performance indicating that GRIL can capture additional features enriching GNNs. We make the complete code for the proposed method available at https://github.com/soham0209/mpml-graph.

翻译：1参数持续同调是拓扑数据分析（TDA）的基石，用于研究隐藏在数据中的连通分量、环等拓扑特征的演化过程。该方法已被应用于增强图神经网络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。为将这些表示整合到机器学习模型中，我们引入了一种名为广义秩不变景观（GRIL）的新型向量表示方法，用于表征2参数持续模。我们证明该向量表示具有1-Lipschitz稳定性，且对底层过滤函数可微，能够便捷地集成到机器学习模型中增强拓扑特征编码。我们提出了一种高效计算该向量表示的算法，并在合成图数据集与基准图数据集上验证方法有效性，与现有1参数和2参数持续模向量表示方法进行对比。进一步，我们将GRIL特征增强到图神经网络中，观察到模型性能提升，这表明GRIL能够捕获额外特征以丰富图神经网络的表征。本文提出方法的完整代码已开源至https://github.com/soham0209/mpml-graph。