Dist2Cycle: A Simplicial Neural Network for Homology Localization
Dist2Cycle: A Simplicial Neural Network for Homology Localization
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DOI：
10.1609/aaai.v36i7.20673
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发表时间：
2021-10
期刊：
影响因子：
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通讯作者：
A. Keros;Vidit Nanda;Kartic Subr
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文献类型：
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作者：
A. Keros;Vidit Nanda;Kartic Subr
单纯形复形可以被看作是图的高维推广，它以不同的分辨率同时对顶点之间的多路有序关系进行显式编码。这个概念是中央对检测数据的高维拓扑特征，特征的图形，编码只成对的关系，保持遗忘。虽然已经尝试将图神经网络（GNN）扩展到单纯复杂设置，但这些方法并不固有地利用或推理网络的底层拓扑结构。我们提出了一个图卷积模型，用于学习由单纯复形的k-同调特征参数化的函数。通过频谱操纵他们的组合k维霍奇拉普拉斯算子，所提出的模型能够学习底层单纯复形的拓扑特征，具体地说，每个k-单纯形与最近的“最佳”第k个同源生成器的距离，有效地提供了一种替代同源定位。
Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations between vertices at different resolutions, all at once. This concept is central towards detection of higher dimensional topological features of data, features to which graphs, encoding only pairwise relationships, remain oblivious. While attempts have been made to extend Graph Neural Networks (GNNs) to a simplicial complex setting, the methods do not inherently exploit, or reason about, the underlying topological structure of the network. We propose a graph convolutional model for learning functions parametrized by the k-homological features of simplicial complexes. By spectrally manipulating their combinatorial k-dimensional Hodge Laplacians, the proposed model enables learning topological features of the underlying simplicial complexes, specifically, the distance of each k-simplex from the nearest "optimal" k-th homology generator, effectively providing an alternative to homology localization.