Metric Reconstruction Via Optimal Transport
Metric Reconstruction Via Optimal Transport
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通过最佳传输重建度量
DOI：
10.1137/17m1148025
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发表时间：
2017
期刊：
影响因子：
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通讯作者：
F. Frick
中科院分区：
文献类型：
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作者：
Michal Adamaszek;Henry Adams;F. Frick
给定度量空间$M$中的点$X$的样本和尺度$r>；0$，Vietoris-Rips单纯复形$\mathm{VR}(X；r)$是试图将$M$从$X$恢复到同伦型的标准构造。这种方法的一个不足之处在于，如果$\mathm{VR}(X；r)$不是局部有限的，则它是不可度量的，因此不能恢复关于$M$的度量信息。为了弥补这一缺陷，我们利用最优传输理论，定义了一个度量空间增厚$X$，我们称之为{Vietoris-Rips增厚}$\m{VR}^m(X；r)$.当$M$是完备黎曼流形，或者是紧致Hadamard空间时，我们证明了Vietoris-Rips加厚满足Hausmann定理($\mathm{vr}^m(M；r)\simeq M$对$r$足够小)，并给出了一个更简单的证明：M$的同伦等价$\mathm{vr}^m(M；r)\，它的同伦逆是(现在连续的)包含映射$M\hookright tarrow\mahorm{vr}^m(M；r)$.此外，我们还描述了在第一个正标度参数$r$处的Vietoris-Rips加厚的同伦型，其中同伦型改变。
Given a sample of points $X$ in a metric space $M$ and a scale $r>0$, the Vietoris-Rips simplicial complex $\mathrm{VR}(X;r)$ is a standard construction to attempt to recover $M$ from $X$ up to homotopy type. A deficiency of this approach is that $\mathrm{VR}(X;r)$ is not metrizable if it is not locally finite, and thus does not recover metric information about $M$. We attempt to remedy this shortcoming by defining a metric space thickening of $X$, which we call the \emph{Vietoris-Rips thickening} $\mathrm{VR}^m(X;r)$, via the theory of optimal transport. When $M$ is a complete Riemannian manifold, or alternatively a compact Hadamard space, we show that the the Vietoris-Rips thickening satisfies Hausmann's theorem ($\mathrm{VR}^m(M;r)\simeq M$ for $r$ sufficiently small) with a simpler proof: homotopy equivalence $\mathrm{VR}^m(M;r)\to M$ is canonically defined as a center of mass map, and its homotopy inverse is the (now continuous) inclusion map $M\hookrightarrow\mathrm{VR}^m(M;r)$. Furthermore, we describe the homotopy type of the Vietoris-Rips thickening of the $n$-sphere at the first positive scale parameter $r$ where the homotopy type changes.