发表机构
King Juan Carlos University; Delft University of Technology（胡安·卡洛斯国王大学; 代尔夫特理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对带噪单纯复形信号，提出基于平滑性和稀疏性假设的非凸优化及块坐标下降算法，以学习单纯复形拓扑，并通过实验验证其有效性。
AI 中文摘要
图是建模复杂数据不规则（非欧几里得）结构的基本工具。然而，它们本质上仅限于表示成对关系，这使得它们不足以处理表现出高阶交互的数据集。单纯复形（SCs）已成为捕获此类高阶依赖关系的有前景框架。本文关注从信号中识别单纯复形拓扑的问题，这构成了基于单纯复形的处理和学习方案的基础。我们考虑一种设置，其中我们观察到与单纯复形节点（0-单纯形）和边子集（1-单纯形）相关联的带噪信号（特征）。我们假设观察到的信号在未知单纯复形拓扑上是平滑的，并且高阶交互是稀疏的。基于这些假设，我们将拓扑学习表述为一个非凸优化问题，并提出一种高效的块坐标下降（BCD）算法来求解。我们表述中的一个关键步骤是使用二元边和三角形选择向量对单纯复形的拓扑进行建模，并结合用于优化此类向量的高效贪心算法。我们建立了收敛到问题松弛（惩罚）版本的平稳点的理论保证，并讨论了计算复杂度。使用合成和真实数据集的多个数值实验验证了我们方法的有效性，突显了单纯复形学习方法在复杂数据集中发现和建模高阶关系的能力。
英文摘要
Graphs are a fundamental tool for modeling the irregular (non-Euclidean) structure of complex data. However, they are inherently limited to representing pairwise relationships, making them inadequate for datasets exhibiting higher-order interactions. Simplicial complexes (SCs) have emerged as a promising framework for capturing such higher-order dependencies. This paper focuses on the problem of identifying the topology of an SC from signals, which serves as the foundation for SC-based processing and learning schemes. We consider a setting where we observe noisy signals (features) associated with the nodes of the SC (0-simplices) and a subset of the edges (1-simplices). We assume the observed signals are smooth over the unknown SC topology, and that the higher-order interactions are sparse. Building on these assumptions, we formulate topology learning as a nonconvex optimization problem and propose an efficient block-coordinate descent (BCD) algorithm to solve it. A key step in our formulation is the modeling of the topology of the SC using binary edge and triangle selection vectors, combined with efficient greedy algorithms for optimizing such vectors. We establish theoretical convergence guarantees to a stationary point of a relaxed (penalized) version of the problem and discuss computational complexity. Multiple numerical experiments with both synthetic and real-world datasets validate the effectiveness of our approach, highlighting the capability of SC-learning methods to uncover and model higher-order relationships in complex datasets.
