34 research outputs found
Convergence between Categorical Representations of Reeb Space and Mapper
The Reeb space, which generalizes the notion of a Reeb graph, is one of the
few tools in topological data analysis and visualization suitable for the study
of multivariate scientific datasets. First introduced by Edelsbrunner et al.,
it compresses the components of the level sets of a multivariate mapping and
obtains a summary representation of their relationships. A related construction
called mapper, and a special case of the mapper construction called the Joint
Contour Net have been shown to be effective in visual analytics. Mapper and JCN
are intuitively regarded as discrete approximations of the Reeb space, however
without formal proofs or approximation guarantees. An open question has been
proposed by Dey et al. as to whether the mapper construction converges to the
Reeb space in the limit.
In this paper, we are interested in developing the theoretical understanding
of the relationship between the Reeb space and its discrete approximations to
support its use in practical data analysis. Using tools from category theory,
we formally prove the convergence between the Reeb space and mapper in terms of
an interleaving distance between their categorical representations. Given a
sequence of refined discretizations, we prove that these approximations
converge to the Reeb space in the interleaving distance; this also helps to
quantify the approximation quality of the discretization at a fixed resolution
Multimapper: Data Density Sensitive Topological Visualization
Mapper is an algorithm that summarizes the topological information contained
in a dataset and provides an insightful visualization. It takes as input a
point cloud which is possibly high-dimensional, a filter function on it and an
open cover on the range of the function. It returns the nerve simplicial
complex of the pullback of the cover. Mapper can be considered a discrete
approximation of the topological construct called Reeb space, as analysed in
the -dimensional case by [Carriere et al.,2018]. Despite its success in
obtaining insights in various fields such as in [Kamruzzaman et al., 2016],
Mapper is an ad hoc technique requiring lots of parameter tuning. There is also
no measure to quantify goodness of the resulting visualization, which often
deviates from the Reeb space in practice. In this paper, we introduce a new
cover selection scheme for data that reduces the obscuration of topological
information at both the computation and visualisation steps. To achieve this,
we replace global scale selection of cover with a scale selection scheme
sensitive to local density of data points. We also propose a method to detect
some deviations in Mapper from Reeb space via computation of persistence
features on the Mapper graph.Comment: Accepted at ICDM
Statistical analysis of Mapper for stochastic and multivariate filters
Reeb spaces, as well as their discretized versions called Mappers, are common
descriptors used in Topological Data Analysis, with plenty of applications in
various fields of science, such as computational biology and data
visualization, among others. The stability and quantification of the rate of
convergence of the Mapper to the Reeb space has been studied a lot in recent
works [BBMW19, CO17, CMO18, MW16], focusing on the case where a scalar-valued
filter is used for the computation of Mapper. On the other hand, much less is
known in the multivariate case, when the codomain of the filter is
, and in the general case, when it is a general metric space , instead of . The few results that are available in this
setting [DMW17, MW16] can only handle continuous topological spaces and cannot
be used as is for finite metric spaces representing data, such as point clouds
and distance matrices. In this article, we introduce a slight modification of
the usual Mapper construction and we give risk bounds for estimating the Reeb
space using this estimator. Our approach applies in particular to the setting
where the filter function used to compute Mapper is also estimated from data,
such as the eigenfunctions of PCA. Our results are given with respect to the
Gromov-Hausdorff distance, computed with specific filter-based pseudometrics
for Mappers and Reeb spaces defined in [DMW17]. We finally provide applications
of this setting in statistics and machine learning for different kinds of
target filters, as well as numerical experiments that demonstrate the relevance
of our approac
Parallel Mapper
The construction of Mapper has emerged in the last decade as a powerful and
effective topological data analysis tool that approximates and generalizes
other topological summaries, such as the Reeb graph, the contour tree, split,
and joint trees. In this paper, we study the parallel analysis of the
construction of Mapper. We give a provably correct parallel algorithm to
execute Mapper on multiple processors and discuss the performance results that
compare our approach to a reference sequential Mapper implementation. We report
the performance experiments that demonstrate the efficiency of our method
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
Structure and Stability of the 1-Dimensional Mapper
Given a continuous function f:X->R and a cover I of its image by intervals, the Mapper is the nerve of a refinement of the pullback cover f^{-1}(I). Despite its success in applications, little is known about the structure and stability of this construction from a theoretical point of view. As a pixelized version of the Reeb graph of f, it is expected to capture a subset of its features (branches, holes), depending on how the interval cover is positioned with respect to the critical values of the function. Its stability should also depend on this positioning. We propose a theoretical framework relating the structure of the Mapper to that of the Reeb graph, making it possible to predict which features will be present and which will be absent in the Mapper given the function and the cover, and for each feature, to quantify its degree of (in-)stability. Using this framework, we can derive guarantees on the structure of the Mapper, on its stability, and on its convergence to the Reeb graph as the granularity of the cover I goes to zero