基于持久同调测度的分形维数
A fractal dimension for measures via persistent homology
摘要：
我们使用持久的同源性，以定义一个家庭的分形维数，表示为$\mathrm{dim}_{\mathrm{PH}}^i(μ)$为每个同调维数$i\ge 0$，分配给一个概率测度$μ$的度量空间。$0$ -维同调（$i=0$）的情况与Michael J. Steele（1988）研究随机抽样点上的最小生成树的总长度的工作有关。事实上，如果$μ$是支持在一个紧凑的子集上的欧几里德空间$\mathbb{R}^m$的$m\ge2$，那么斯蒂尔的工作意味着，如果绝对连续部分的$μ$有积极的质量，$\mathrm{dim}_{\mathrm{PH}}^0(μ)=m$，否则$\mathrm{dim}_{\mathrm{PH}}^0(μ)<m$。实验表明，类似的结果可能是正确的高维同源$0<i<m$，虽然这是一个悬而未决的问题。我们的分形维数是通过考虑一个极限来定义的，当点$n$的数量趋于无穷大时，从i.i.d.中的$μ$中选择的$n$随机点的$i$维持久同源区间长度的总和。时尚.对于某些措施$μ,$，我们能够分配一个更精细的不变量，一条曲线测量的限制分布持久的同源区间长度的点的数量趋于无穷大。证明了当$μ$是单位区间上的均匀分布时，在$0$维同调的情形下，该极限曲线存在;并猜想当$μ$是欧氏空间中紧致集的重标度概率测度且Lebesgue测度为正时，该极限曲线存在.
英文摘要：
We use persistent homology in order to define a family of fractal dimensions, denoted $\mathrm{dim}_{\mathrm{PH}}^i(μ)$ for each homological dimension $i\ge 0$, assigned to a probability measure $μ$ on a metric space. The case of $0$-dimensional homology ($i=0$) relates to work by Michael J Steele (1988) studying the total length of a minimal spanning tree on a random sampling of points. Indeed, if $μ$ is supported on a compact subset of Euclidean space $\mathbb{R}^m$ for $m\ge2$, then Steele's work implies that $\mathrm{dim}_{\mathrm{PH}}^0(μ)=m$ if the absolutely continuous part of $μ$ has positive mass, and otherwise $\mathrm{dim}_{\mathrm{PH}}^0(μ)<m$. Experiments suggest that similar results may be true for higher-dimensional homology $0<i<m$, though this is an open question. Our fractal dimension is defined by considering a limit, as the number of points $n$ goes to infinity, of the total sum of the $i$-dimensional persistent homology interval lengths for $n$ random points selected from $μ$ in an i.i.d. fashion. To some measures $μ,$ we are able to assign a finer invariant, a curve measuring the limiting distribution of persistent homology interval lengths as the number of points goes to infinity. We prove this limiting curve exists in the case of $0$-dimensional homology when $μ$ is the uniform distribution over the unit interval, and conjecture that it exists when $μ$ is the rescaled probability measure for a compact set in Euclidean space with positive Lebesgue measure.
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