A family of metrics from the truncated smoothing of Reeb graphs
A family of metrics from the truncated smoothing of Reeb graphs
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DOI：
10.4230/lipics.socg.2021.22
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发表时间：
2020-07
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影响因子：
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通讯作者：
E. Chambers;E. Munch;Tim Ophelders
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文献类型：
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作者：
E. Chambers;E. Munch;Tim Ophelders
在本文中，我们引入了一个扩展的平滑Reeb图，我们称之为截断平滑，这反过来又允许我们定义一个新的家庭的度量，推广的交织距离Reeb图。直观地说，我们在平滑过程中“砍掉”了局部最小值和最大值附近的部分，其中切割量由参数$\tau$控制。在将截断形式化为一个函子之后，我们证明了当在平滑函子之后应用时，这可以防止函数范围的广泛扩展，并且当与$0 \leq \tau \leq 2\varepsilon$的平滑相结合时，会产生特别好的特性（例如保持连通性），其中$\varepsilon$是平滑参数。然后，对于$\tau \in [0，\varepaly]$的限制，我们有额外的结构，我们可以利用它来构造一个类别流，用于任何斜率$m \in [0，1]$的选择。使用为具有流的类别构建的基础设施，这然后给出了每个$m \in [0，1]$的交织距离，这是原始交织距离的推广，这是$m=0$的情况。虽然由此产生的度量是不稳定的，我们表明，任何对这些为$m，m' \在[0，1）$是强等价的度量，这反过来又给每个度量的稳定性，直到一个乘法常数。最后，我们讨论的影响，这个指标在更广泛的家庭指标Reeb图。
In this paper, we introduce an extension of smoothing on Reeb graphs, which we call truncated smoothing; this in turn allows us to define a new family of metrics which generalize the interleaving distance for Reeb graphs. Intuitively, we "chop off" parts near local minima and maxima during the course of smoothing, where the amount cut is controlled by a parameter $\tau$. After formalizing truncation as a functor, we show that when applied after the smoothing functor, this prevents extensive expansion of the range of the function, and yields particularly nice properties (such as maintaining connectivity) when combined with smoothing for $0 \leq \tau \leq 2\varepsilon$, where $\varepsilon$ is the smoothing parameter. Then, for the restriction of $\tau \in [0,\varepsilon]$, we have additional structure which we can take advantage of to construct a categorical flow for any choice of slope $m \in [0,1]$. Using the infrastructure built for a category with a flow, this then gives an interleaving distance for every $m \in [0,1]$, which is a generalization of the original interleaving distance, which is the case $m=0$. While the resulting metrics are not stable, we show that any pair of these for $m,m' \in [0,1)$ are strongly equivalent metrics, which in turn gives stability of each metric up to a multiplicative constant. We conclude by discussing implications of this metric within the broader family of metrics for Reeb graphs.