Abstract
Abstract
We study the probabilistic convergence between the mapper graph and the Reeb graph of a topological space $${\mathbb {X}}$$
X
equipped with a continuous function $$f: {\mathbb {X}}\rightarrow \mathbb {R}$$
f
:
X
→
R
. 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 $$\mathbb {R}$$
R
. We then introduce a variant of the classic mapper graph of Singh et al. (in: Eurographics symposium on point-based graphics, 2007), referred to as the enhanced mapper graph, and demonstrate that such a construction approximates the Reeb graph of $$({\mathbb {X}}, f)$$
(
X
,
f
)
when it is applied to points randomly sampled from a probability density function concentrated on $$({\mathbb {X}}, f)$$
(
X
,
f
)
. Our techniques are based on the interleaving distance of constructible cosheaves and topological estimation via kernel density estimates. Following Munch and Wang (In: 32nd international symposium on computational geometry, volume 51 of Leibniz international proceedings in informatics (LIPIcs), Dagstuhl, Germany, pp 53:1–53:16, 2016), we first show that the mapper graph of $$({\mathbb {X}}, f)$$
(
X
,
f
)
, a constructible $$\mathbb {R}$$
R
-space (with a fixed open cover), approximates the Reeb graph of the same space. We then construct an isomorphism between the mapper of $$({\mathbb {X}},f)$$
(
X
,
f
)
to the mapper of a super-level set of a probability density function concentrated on $$({\mathbb {X}}, f)$$
(
X
,
f
)
. Finally, building on the approach of Bobrowski et al. (Bernoulli 23(1):288–328, 2017b), 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.