拓扑盲点：通过表达能力视角理解与扩展拓扑深度学习
Topological Blindspots: Understanding and Extending Topological Deep Learning Through the Lens of Expressivity
AI总结：
针对拓扑深度学习（TDL）的高阶消息传递（HOMP）框架存在拓扑与度量不变量捕捉能力不足等问题，提出多细胞网络（MCN）及可扩展版本SMCN，新架构在基准与真实数据集上表现优于现有方法，为TDL表达能力研究提供新视角与解决方案。
摘要：
拓扑深度学习（TDL）是一个快速发展的领域，旨在利用数据中的拓扑结构，促进从以拓扑对象为支撑的数据（从分子到3D形状）中学习。大多数TDL架构可在高阶消息传递（HOMP）框架下统一，该框架将图消息传递推广到高阶域。在论文第一部分，我们从拓扑视角探索HOMP的表达能力，证明该框架无法捕捉直径、可定向性、平面性和同调等基本拓扑与度量不变量。此外，我们展示HOMP在图上充分利用提升和池化方法的局限性。据我们所知，这是首个从拓扑视角研究TDL表达能力的工作。在论文第二部分，我们开发两类新架构——多细胞网络（MCN）和可扩展MCN（SMCN）——其灵感来自具有表达能力的图神经网络（GNN）。MCN可实现完全表达能力，但将其扩展到大型数据对象在计算上成本高昂。SMCN作为更具可扩展性的替代方案，仍能缓解HOMP的许多表达能力局限性。最后，我们基于模型学习复形拓扑性质的能力创建新基准，然后在这些基准和真实世界图数据集上评估SMCN，证明其优于HOMP基线和具有表达能力的图方法，凸显了富有表达力地利用拓扑信息的价值。代码和数据可在https://github.com/yoavgelberg/SMCN获取。
英文摘要：
Topological deep learning (TDL) is a rapidly growing field that seeks to leverage topological structure in data and facilitate learning from data supported on topological objects, ranging from molecules to 3D shapes. Most TDL architectures can be unified under the framework of higher-order message-passing (HOMP), which generalizes graph message-passing to higher-order domains. In the first part of the paper, we explore HOMP's expressive power from a topological perspective, demonstrating the framework's inability to capture fundamental topological and metric invariants such as diameter, orientability, planarity, and homology. In addition, we demonstrate HOMP's limitations in fully leveraging lifting and pooling methods on graphs. To the best of our knowledge, this is the first work to study the expressivity of TDL from a \emph{topological} perspective. In the second part of the paper, we develop two new classes of architectures -- multi-cellular networks (MCN) and scalable MCN (SMCN) -- which draw inspiration from expressive GNNs. MCN can reach full expressivity, but scaling it to large data objects can be computationally expansive. Designed as a more scalable alternative, SMCN still mitigates many of HOMP's expressivity limitations. Finally, we create new benchmarks for evaluating models based on their ability to learn topological properties of complexes. We then evaluate SMCN on these benchmarks and on real-world graph datasets, demonstrating improvements over both HOMP baselines and expressive graph methods, highlighting the value of expressively leveraging topological information. Code and data are available at https://github.com/yoavgelberg/SMCN.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
