Motivation: The Mapper algorithm is an essential tool to explore shape of
data in topology data analysis. With a dataset as an input, the Mapper
algorithm outputs a graph representing the topological features of the whole
dataset. This graph is often regarded as an approximation of a reeb graph of
data. The classic Mapper algorithm uses fixed interval lengths and overlapping
ratios, which might fail to reveal subtle features of data, especially when the
underlying structure is complex.
Results: In this work, we introduce a distribution guided Mapper algorithm
named D-Mapper, that utilizes the property of the probability model and data
intrinsic characteristics to generate density guided covers and provides
enhanced topological features. Our proposed algorithm is a probabilistic
model-based approach, which could serve as an alternative to non-prababilistic
ones. Moreover, we introduce a metric accounting for both the quality of
overlap clustering and extended persistence homology to measure the performance
of Mapper type algorithm. Our numerical experiments indicate that the D-Mapper
outperforms the classical Mapper algorithm in various scenarios. We also apply
the D-Mapper to a SARS-COV-2 coronavirus RNA sequences dataset to explore the
topological structure of different virus variants. The results indicate that
the D-Mapper algorithm can reveal both vertical and horizontal evolution
processes of the viruses.
Availability: Our package is available at
this https URL

基于概率模型和数据特征的 D-Mapper 算法可以生成密度引导覆盖，并提供增强的拓扑特征，对比传统 Mapper 算法表现更好，同时可用于探索不同病毒变体的拓扑结构。

一种基于分布的 Mapper 算法

A distribution-guided Mapper algorithm

The Mapper algorithm is a visualization technique in topological data
analysis (TDA) that outputs a graph reflecting the structure of a given
dataset. The Mapper algorithm requires tuning several parameters in order to
generate a "nice" Mapper graph. The paper focuses on selecting the cover
parameter. We present an algorithm that optimizes the cover of a Mapper graph
by splitting a cover repeatedly according to a statistical test for normality.
Our algorithm is based on $G$-means clustering which searches for the optimal
number of clusters in $k$-means by conducting iteratively the Anderson-Darling
test. Our splitting procedure employs a Gaussian mixture model in order to
choose carefully the cover based on the distribution of a given data.
Experiments for synthetic and real-world datasets demonstrate that our
algorithm generates covers so that the Mapper graphs retain the essence of the
datasets.

本文介绍了优化 Mapper 图的覆盖参数的算法，该算法基于 G-means 聚类，并使用高斯混合模型选择适当的覆盖，实验证明该算法生成的 Mapper 图保留了数据集的本质。

$G$-Mapper: 在 Mapper 构造中学习封面

$G$-Mapper: Learning a Cover in the Mapper Construction

The mapper algorithm is a popular tool from topological data analysis for
extracting topological summaries of high-dimensional datasets. In this paper,
we present Mapper Interactive, a web-based framework for the interactive
analysis and visualization of high-dimensional point cloud data. It implements
the mapper algorithm in an interactive, scalable, and easily extendable way,
thus supporting practical data analysis. In particular, its command-line API
can compute mapper graphs for 1 million points of 256 dimensions in about 3
minutes (4 times faster than the vanilla implementation). Its visual interface
allows on-the-fly computation and manipulation of the mapper graph based on
user-specified parameters and supports the addition of new analysis modules
with a few lines of code. Mapper Interactive makes the mapper algorithm
accessible to nonspecialists and accelerates topological analytics workflows.

本文介绍了基于 Web 框架的交互式分析和可视化高维点云数据的 Mapper Interactive 算法，该方法实现了映射算法的交互式、可扩展和可扩展性，支持实践数据分析和加速拓扑分析工作流程。

Mapper Interactive：高维数据可视化的可扩展交互式工具箱

Mapper Interactive: A Scalable, Extendable, and Interactive Toolbox for  the Visual Exploration of High-Dimensional Data

We describe Fibres of Failure (FiFa), a method to classify failure modes of
predictive processes using the Mapper algorithm from Topological Data Analysis.
Our method uses Mapper to build a graph model of input data stratified by
prediction error.
Groupings found in high-error regions of the Mapper model then provide
distinct failure modes of the predictive process.
We demonstrate FiFa on misclassifications of MNIST images with added noise,
and demonstrate two ways to use the failure mode classification: either to
produce a correction layer that adjusts predictions by similarity to the
failure modes; or to inspect members of the failure modes to illustrate and
investigate what characterizes each failure mode.

使用拓扑数据分析中的 Mapper 算法，本文提出了一种名为 FiFa 的方法，用于分类预测过程的失败模式。在 MNIST 图像添加噪声的情况下，通过对高误差区域的群集分析，可得到预测过程的区别失败模式。本文还演示了两种使用失败模式分类的方法：一是产生一层校正层以通过失败模式的相似性来调整预测，另一种方法是检查失败模式的成员，以说明并调查每种失败模式的特点。