循环图的持久同调
The persistent homology of cyclic graphs
摘要：
给出了一个计算循环图的团复形在$n$顶点上的滤除的$k$维持久同调的$O(n^2(k+\log n))$算法.这在顶点数$n$上几乎是二次的，因此对传统的持续同调算法有很大的改进，传统的持续同调算法在维数至多为$k+1$的单形数上是三次的，因此在顶点数$n$上运行时间$O(n^{3(k+2)})$是三次的。我们的算法适用于，例如，Vietoris-裂口复杂的采样点从曲线在$\mathbb{R}^d$的规模是有界的曲线的几何形状，但仍然足够大，使Vietoris-裂口复杂的可能有非平凡的同源性在任意高的维度$k$。在平面$\mathbb{R}^2$的情况下，我们证明了我们的算法适用于所有的尺度参数，如果$n$顶点是从一个凸闭可微曲线，其凸包包含其渐屈线采样。我们问，是否有其他几何设置中，计算持久的同源性是（说）二次或三次的顶点数，而不是在单形的数量。
英文摘要：
We give an $O(n^2(k+\log n))$ algorithm for computing the $k$-dimensional persistent homology of a filtration of clique complexes of cyclic graphs on $n$ vertices. This is nearly quadratic in the number of vertices $n$, and therefore a large improvement upon the traditional persistent homology algorithm, which is cubic in the number of simplices of dimension at most $k+1$, and hence of running time $O(n^{3(k+2)})$ in the number of vertices $n$. Our algorithm applies, for example, to Vietoris--Rips complexes of points sampled from a curve in $\mathbb{R}^d$ when the scale is bounded depending on the geometry of the curve, but still large enough so that the Vietoris--Rips complex may have non-trivial homology in arbitrarily high dimensions $k$. In the case of the plane $\mathbb{R}^2$, we prove that our algorithm applies for all scale parameters if the $n$ vertices are sampled from a convex closed differentiable curve whose convex hull contains its evolute. We ask if there are other geometric settings in which computing persistent homology is (say) quadratic or cubic in the number of vertices, instead of in the number of simplices.
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