GRIL: A 2-parameter Persistence Based Vectorization for Machine Learning
GRIL: A 2-parameter Persistence Based Vectorization for Machine Learning
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DOI：
10.48550/arxiv.2304.04970
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发表时间：
2023-04
期刊：
影响因子：
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通讯作者：
Cheng Xin;Soham Mukherjee;Shreyas N. Samaga;T. Dey
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文献类型：
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作者：
Cheng Xin;Soham Mukherjee;Shreyas N. Samaga;T. Dey
$ 1 $ - 参数持续的同源性，拓扑数据分析（TDA）的基石（TDA）研究了拓扑特征的演变，例如隐藏在数据中的连接组件和周期。它已用于增强深度学习模型的表示能力，例如图神经网络（GNNS）。为了丰富拓扑特征的表示，我们在这里建议研究由双过滤功能引起的$ 2 $参数持久模块。为了将这些表示形式纳入机器学习模型，我们引入了一种新颖的向量表示，称为广义等级不变景观（GRIL），价格为$ 2 $ - 参数持久性模块。我们表明，相对于基础过滤功能，该矢量表示为$ 1 $ -LIPSCHITZ稳定且可区分，并且可以轻松地集成到机器学习模型中，以增强编码拓扑功能的编码。我们提出了一种有效计算矢量表示的算法。我们还测试了关于合成和基准图数据集的方法，并将结果与先前的矢量表示形式进行比较$ 1 $ - 参数和$ 2 $ - 参数持续模块。此外，我们以Gril特征增强GNN，并观察到性能的提高，表明Gril可以捕获富含GNN的其他功能。我们在https://github.com/soham0209/mpml-graph上为提出的方法提供完整的代码。
$1$-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.