通过线性插值的锯齿形持续同调的可微性框架
A differentiability framework for zigzag persistent homology via linear interpolation
- Area Science Park（的里雅斯特科技园）
- Politecnico di Torino（都灵理工大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出了锯齿形持续同调的可微性框架，通过线性插值将滤波值连续性传递给图点，并在排除集外证明局部Lipschitz连续性，实验验证了其在传感器网络覆盖优化和动态图分类中的有效性。
AI中文摘要：
持续同调可以被微分并整合到学习流程中，但对于锯齿形持续同调，尚不存在类似的框架，而后者在底层拓扑结构随时间非单调演化时是必需的。我们为通过固定复形上的时变滤波值阈值化得到的单纯复形序列开发了这样的框架。通过将持续同调图端点赋予线性插值滤波值穿越阈值的实值时间，我们将滤波值的连续性转移到图点上。这产生了所得持续同调图值映射的光滑局部提升，由此我们可以在滤波值参数化的温和正则性条件下几乎处处推导微分。我们证明了在显式的零测度排除集之外具有局部Lipschitz连续性；因此，标准随机次梯度收敛保证并不直接适用。我们认为，即使没有这样的保证，该排除集在实践中也足够小，以允许有效的优化。我们在两个实验中对此进行了实证测试：传感器网络覆盖优化和动态图分类。
英文摘要：
Persistent homology can be differentiated and incorporated into learning pipelines, but no analogous framework exists for zigzag persistence, which is needed when the underlying topological structure evolves non-monotonically over time. We develop such a framework for sequences of simplicial complexes obtained by thresholding time-dependent filtering values on a fixed complex. By assigning persistence diagram endpoints the real-valued times at which linearly interpolated filtering values cross the threshold, we transfer the continuity of the filtering values to the diagram points. This yields smooth local lifts of the resulting persistence-diagram-valued map, from which we derive differentials almost everywhere under mild regularity conditions on the parametrization of the filtering values. We prove local Lipschitz continuity outside an explicit measure-zero exclusion set; standard stochastic subgradient convergence guarantees therefore do not apply directly. We argue that, even without such guarantees, this exclusion set is small enough in practice to allow effective optimization. We test this empirically in two experiments: sensor network coverage optimization and dynamic graph classification.
