Billera-Holmes-Vogtmann树空间中Frechet平均的性质
Properties for the Frechet Mean in Billera-Holmes-Vogtmann Treespace
摘要：
加权树的Billera-Holmes-Vogtmann（BHV）空间可以嵌入到欧几里得空间中，但外欧式均值往往位于树空间之外。Sturm证明了内在Frechet均值在树空间中是存在且唯一的。这个Frechet平均值可以用迭代算法近似，但是算法收敛的界限是未知的，并且没有其他已知的多项式算法用于计算Frechet平均值，甚至也没有平均值中存在的边缘。给出了边在Frechet平均中的第一个充要条件。这些条件以边的权的不等式的形式存在。这些条件提供了一个预处理步骤，用于找到包含Frechet均值的树空间正交。这项工作推广到orthant空间。
英文摘要：
The Billera-Holmes-Vogtmann (BHV) space of weighted trees can be embedded in Euclidean space, but the extrinsic Euclidean mean often lies outside of treespace. Sturm showed that the intrinsic Frechet mean exists and is unique in treespace. This Frechet mean can be approximated with an iterative algorithm, but bounds on the convergence of the algorithm are not known, and there is no other known polynomial algorithm for computing the Frechet mean nor even the edges present in the mean. We give the first necessary and sufficient conditions for an edge to be in the Frechet mean. The conditions are in the form of inequalities on the weights of the edges. These conditions provide a pre-processing step for finding the treespace orthant containing the Frechet mean. This work generalizes to orthant spaces.
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