Probabilistic Fréchet means for time varying persistence diagrams
Probabilistic Fréchet means for time varying persistence diagrams
复制标题
概率 Fréchet 表示随时间变化的持久性图
DOI：
10.1214/15-ejs1030
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发表时间：
2013
影响因子：
1.1
通讯作者：
J. Harer
中科院分区：
文献类型：
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作者：
E. Munch;Katharine Turner;Paul Bendich;S. Mukherjee;J. Mattingly;J. Harer
为了使用持久性图作为一个真正的统计工具，对一组图的均值和方差有一个好的概念是非常有用的。2011年，Mileyko和他的合作者首次研究了$（\mathcal{D}_p，W_p）$中Fr\'echet平均的性质，该空间是配备了第p个Wasserstein度量的持久性图空间。特别是，他们表明，一个有限的图集的Fr\'echet平均值总是存在的，但不一定是唯一的。一组连续变化的图表的平均值本身并不（一定）连续变化，这在试图将Fr\'echet平均值定义扩展到葡萄园领域时会出现明显的问题。
我们通过改变Fr 'echet平均值的原始定义来解决这个问题，使其现在成为持久性图集合上的概率度量;简而言之，一组图的平均值将是原子度量的加权和，其中每个原子本身就是使用输入图的扰动确定的持久性图。这个定义为每个$N$给出了一个映射$（\mathcal{D}_p）^N \to \mathbb{P}（\mathcal{D}_p）$。我们证明了这个映射在有限图上是H\“older连续的，因此可以用来建立一个有用的统计量时变持久性图，更好地称为葡萄园。
In order to use persistence diagrams as a true statistical tool, it would be very useful to have a good notion of mean and variance for a set of diagrams. In 2011, Mileyko and his collaborators made the first study of the properties of the Fr\'echet mean in $(\mathcal{D}_p,W_p)$, the space of persistence diagrams equipped with the p-th Wasserstein metric. In particular, they showed that the Fr\'echet mean of a finite set of diagrams always exists, but is not necessarily unique. The means of a continuously-varying set of diagrams do not themselves (necessarily) vary continuously, which presents obvious problems when trying to extend the Fr\'echet mean definition to the realm of vineyards.
We fix this problem by altering the original definition of Fr\'echet mean so that it now becomes a probability measure on the set of persistence diagrams; in a nutshell, the mean of a set of diagrams will be a weighted sum of atomic measures, where each atom is itself a persistence diagram determined using a perturbation of the input diagrams. This definition gives for each $N$ a map $(\mathcal{D}_p)^N \to \mathbb{P}(\mathcal{D}_p)$. We show that this map is H\"older continuous on finite diagrams and thus can be used to build a useful statistic on time-varying persistence diagrams, better known as vineyards.
DOI：
--
发表时间：
2022
期刊：
影响因子：
--
作者：
Ulrich Aivodji;Hiromi Arai;Sebastien Gambs;Satoshi Hara;Tam Le
通讯作者：
Tam Le