The homology groups of a simplicial complex reveal fundamental properties of the topology of the data or the system and the notion of topological stability naturally poses an important yet not fully investigated question. In the current work, we study the stability in terms of the smallest perturbation sufficient to change the dimensionality of the corresponding homology group. Such definition requires an appropriate weighting and normalizing procedure for the boundary operators acting on the Hodge algebra's homology groups. Using the resulting boundary operators, we then formulate the question of structural stability as a spectral matrix nearness problem for the corresponding higher-order graph Laplacian. We develop a bilevel optimization procedure suitable for the formulated matrix nearness problem and illustrate the method's performance on a variety of synthetic quasi-triangulation datasets and transportation networks.

翻译：单纯复形的同调群揭示了数据或系统拓扑的基本性质，而拓扑稳定性的概念自然地引出一个重要但尚未被充分研究的问题。在本文中，我们从最小扰动足以改变相应同调群维度的角度研究稳定性。该定义需要对作用于霍奇代数同调群的边界算子进行适当的加权与归一化处理。基于得到的边界算子，我们将结构稳定性问题转化为对应高阶图拉普拉斯算子的谱矩阵逼近问题。我们发展了一种适用于该矩阵逼近问题的双层优化方法，并通过多种合成拟三角剖分数据集与交通网络验证了该方法的性能。