Title: An introduction to Topological Data Analysis: fundamental and practical aspects for data scientists,
Abstract: Topological Data Analysis (tda) is a recent and fast growing eld providing a
set of new topological and geometric tools to infer relevant features for
possibly complex data. This paper is a brief introduction, through a few
selected topics, to basic fundamental and practical aspects of tda for non
experts. 1 Introduction and motivation Topological Data Analysis (tda) is a
recent eld that emerged from various works in applied (algebraic) topology and
computational geometry during the rst decade of the century. Although one can
trace back geometric approaches for data analysis quite far in the past, tda
really started as a eld with the pioneering works of Edelsbrunner et al. (2002)
and Zomorodian and Carlsson (2005) in persistent homology and was popularized
in a landmark paper in 2009 Carlsson (2009). tda is mainly motivated by the
idea that topology and geometry provide a powerful approach to infer robust
qualitative, and sometimes quantitative, information about the structure of
data-see, e.g. Chazal (2017). tda aims at providing well-founded mathematical,
statistical and algorithmic methods to infer, analyze and exploit the complex
topological and geometric structures underlying data that are often represented
as point clouds in Euclidean or more general metric spaces. During the last few
years, a considerable eort has been made to provide robust and ecient data
structures and algorithms for tda that are now implemented and available and
easy to use through standard libraries such as the Gudhi library (C++ and
Python) Maria et al. (2014) and its R software interface Fasy et al. (2014a).
Although it is still rapidly evolving, tda now provides a set of mature and
ecient tools that can be used in combination or complementary to other data
sciences tools. The tdapipeline. tda has recently known developments in various
directions and application elds. There now exist a large variety of methods
inspired by topological and geometric approaches. Providing a complete overview
of all these existing approaches is beyond the scope of this introductory
survey. However, most of them rely on the following basic and standard pipeline
that will serve as the backbone of this paper: 1. The input is assumed to be a
nite set of points coming with a notion of distance-or similarity between them.
This distance can be induced by the metric in the ambient space (e.g. the
Euclidean metric when the data are embedded in R d) or come as an intrinsic
metric dened by a pairwise distance matrix. The denition of the metric on the
data is usually given as an input or guided by the application. It is however
important to notice that the choice of the metric may be critical to reveal
interesting topological and geometric features of the data.