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Approximations of 1-Dimensional Intrinsic Persistence of Geodesic Spaces and Their Stability. (arXiv:1711.05111v2 [math.GT] UPDATED)

来源于:arXiv
A standard way of approximating or discretizing a metric space is by taking its Rips complexes. These approximations for all parameters are often bound together into a filtration, to which we apply the fundamental group or the first homology. We call the resulting object persistence. Recent results demonstrate that persistence of a compact geodesic locally contractible space $X$ carries a lot of geometric information. However, by definition the corresponding Rips complexes have uncountably many vertices. In this paper we show that nonetheless, the whole persistence of $X$ may be obtained by an appropriate finite sample (subset of $X$), and that persistence of any subset of $X$ is well interleaved with the persistence of $X$. It follows that the persistence of $X$ is the minimum of persistences obtained by all finite samples. Furthermore, we prove a much improved Stability theorem for such approximations. As a special case we provide for each $r>0$ a density $s>0$, so that for eac 查看全文>>