Probabilistic convergence and stability of random mapper graphs

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.

An Introduction to Topological Data Analysis: Fundamental and Practical Aspects for Data Scientists

“…Each time series is thus described by 6,000 topological variables. To predict the walker from these features, we use a random forest ( Breiman, 2001 ), which is known to be efficient in such a high-dimensional setting. We split the data into train and test samples at random several times.…”

“…The case of stochastic and multivariate filters has also been studied by Carrière and Michel (2019). An alternative description of the probabilistic convergence of Mapper, in terms of categorification, has also been proposed in the study by Brown et al (2020). Other approaches have been proposed to study and deal with the (left).…”

Scalar Field Comparison with Topological Descriptors: Properties and Applications for Scientific Visualization

“…One approach to efficiency is to develop approximation algorithms for computing the comparative measures. This approach can be achieved by reducing the complexity of the input data using concepts proposed in mapper [SMC07] or cosheaf [DSMP16, BBMW21] w.r.t . the Reeb graph, which are gaining popularity in computational topology.…”

“…The extended Reeb graph [BB14] uses cover elements from a partition of the domain without overlaps. The enhanced mapper graph [BBMW21] considers inverse images of intersections among the cover elements and encodes function values on its vertices and edges. Several variants of mapper constructions exist, as discussed in Sect.…”

A Gradient Sampling Algorithm for Stratified Maps with Applications to Topological Data Analysis

“…For instance, the branches of the Mapper graph in [65] correspond to the differentiation of stem cells into specialized cells. Besides its potential for applications, Mapper enjoys strong statistical and topological properties [12,21,20,19,59,6].…”