4 research outputs found
Local Equivalence and Intrinsic Metrics between Reeb Graphs
As graphical summaries for topological spaces and maps, Reeb graphs are
common objects in the computer graphics or topological data analysis
literature. Defining good metrics between these objects has become an important
question for applications, where it matters to quantify the extent by which two
given Reeb graphs differ. Recent contributions emphasize this aspect, proposing
novel distances such as {\em functional distortion} or {\em interleaving} that
are provably more discriminative than the so-called {\em bottleneck distance},
being true metrics whereas the latter is only a pseudo-metric. Their main
drawback compared to the bottleneck distance is to be comparatively hard (if at
all possible) to evaluate. Here we take the opposite view on the problem and
show that the bottleneck distance is in fact good enough {\em locally}, in the
sense that it is able to discriminate a Reeb graph from any other Reeb graph in
a small enough neighborhood, as efficiently as the other metrics do. This
suggests considering the {\em intrinsic metrics} induced by these distances,
which turn out to be all {\em globally} equivalent. This novel viewpoint on the
study of Reeb graphs has a potential impact on applications, where one may not
only be interested in discriminating between data but also in interpolating
between them
Probabilistic Convergence and Stability of Random Mapper Graphs
We study the probabilistic convergence between the mapper graph and the Reeb
graph of a topological space equipped with a continuous function
. We first give a categorification of the
mapper graph and the Reeb graph by interpreting them in terms of cosheaves and
stratified covers of the real line . We then introduce a variant of
the classic mapper graph of Singh et al.~(2007), referred to as the enhanced
mapper graph, and demonstrate that such a construction approximates the Reeb
graph of when it is applied to points randomly sampled from a
probability density function concentrated on .
Our techniques are based on the interleaving distance of constructible
cosheaves and topological estimation via kernel density estimates. Following
Munch and Wang (2018), we first show that the mapper graph of , a constructible -space (with a fixed open cover), approximates
the Reeb graph of the same space. We then construct an isomorphism between the
mapper of to the mapper of a super-level set of a probability
density function concentrated on . Finally, building on the
approach of Bobrowski et al.~(2017), we show that, with high probability, we
can recover the mapper of the super-level set given a sufficiently large
sample. Our work is the first to consider the mapper construction using the
theory of cosheaves in a probabilistic setting. It is part of an ongoing effort
to combine sheaf theory, probability, and statistics, to support topological
data analysis with random data