[论文解读] Topological Deep Learning: Going Beyond Graph Data

Mustafa Hajij, Ghada Zamzmi+12 more|arXiv (Cornell University)|Jun 1, 2022

Topological and Geometric Data Analysis被引用 26

一句话总结

本论文将组合复合体（CCs）作为统一的拓扑领域引入，发展基于 push-forward 的高阶消息传递的 CCNNs，并在形状分析和图学习任务中展示出具有竞争力的性能。

ABSTRACT

Topological deep learning is a rapidly growing field that pertains to the development of deep learning models for data supported on topological domains such as simplicial complexes, cell complexes, and hypergraphs, which generalize many domains encountered in scientific computations. In this paper, we present a unifying deep learning framework built upon a richer data structure that includes widely adopted topological domains. Specifically, we first introduce combinatorial complexes, a novel type of topological domain. Combinatorial complexes can be seen as generalizations of graphs that maintain certain desirable properties. Similar to hypergraphs, combinatorial complexes impose no constraints on the set of relations. In addition, combinatorial complexes permit the construction of hierarchical higher-order relations, analogous to those found in simplicial and cell complexes. Thus, combinatorial complexes generalize and combine useful traits of both hypergraphs and cell complexes, which have emerged as two promising abstractions that facilitate the generalization of graph neural networks to topological spaces. Second, building upon combinatorial complexes and their rich combinatorial and algebraic structure, we develop a general class of message-passing combinatorial complex neural networks (CCNNs), focusing primarily on attention-based CCNNs. We characterize permutation and orientation equivariances of CCNNs, and discuss pooling and unpooling operations within CCNNs in detail. Third, we evaluate the performance of CCNNs on tasks related to mesh shape analysis and graph learning. Our experiments demonstrate that CCNNs have competitive performance as compared to state-of-the-art deep learning models specifically tailored to the same tasks. Our findings demonstrate the advantages of incorporating higher-order relations into deep learning models in different applications.

研究动机与目标

• 将组合复合体（CCs）作为一个统一的拓扑领域，引入它以概括图、单体/单胞复合体和超图。
• 发展一种通用的组合复合神经网络（CCNNs）类，包括卷积、注意力和池化/反池化 运算。
• 在 CCNN 框架中定义并分析高阶消息传递、等变性以及池化。
• 提供在网格形状分析和图学习方面的实际实现与基准测试，以证明其有效性。

提出的方法

• 将 CCs 定义为具有分层集合关系的通用拓扑域。
• 开发以 push-forward 运算为基础的 CCNNs，能够实现高阶消息传递。
• 将 CC 卷积、CC 注意力、以及 CC 池化/反池化 表述为张量运算。
• 确立 CCNNs 的置换和方向等变性，并将 CCNNs 与 Hasse 图表示联系起来。
• 提供软件库（TopoNetX、TopoEmbedX、TopoModelX）和用于形状/机器学习任务的实验流程。

实验结果

研究问题

• RQ1 组合复合体（CCs）能否作为学习于图、单体/单胞复合体和超图的统一域？
• RQ2 如何通过 push-forward 运算在 CCNNs 中实现有效的高阶消息传递？
• RQ3 CCNNs 的等变性性质（置换、方向）是什么，在该框架下池化/反池化运算的行为如何？
• RQ4 与最先进模型相比，CCNNs 在网格形状分析和图学习任务中是否能达到具有竞争力的性能？

主要发现

• CCNNs 为跨高阶域的拓扑深度学习提供了统一的蓝本。
• 基于 push-forward 的消息传递的 CCNNs 在网格分割/分类和图任务上取得了具有竞争力的性能。
• 高阶关系以及 CC 池化/反池化能够在 CC 维度之间鲁棒地提升信号，实现长距离信息传播。
• 该框架与拓扑数据分析中的 mapper 构造相关，并展示了如何将 CCs 简化为用于图分析的 Hasse 图。