Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A family of metrics from the truncated smoothing of Reeb graphs   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.

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The categories of this paper are: mtrl-th
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The categories of this paper are: cmp-lg
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The categories of this paper are: cond-mat
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The categories of this paper are: eess
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The categories of this paper are: comp-gas
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The categories of this paper are: adap-org
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The categories of this paper are: hep-th
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The categories of this paper are: nucl-th
0
The categories of this paper are: hep-ph
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The categories of this paper are: q-bio
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The categories of this paper are: bayes-an
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The categories of this paper are: funct-an
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The categories of this paper are: stat
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The categories of this paper are: chao-dyn
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The categories of this paper are: q-fin
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The categories of this paper are: plasm-ph