With the recent explosion in the amount, the variety, and the dimensionality of available data, identifying, extracting, and exploiting their underlying structure has become a problem of fundamental importance for data analysis and statistical learning. Topological data analysis tda is a recent and fast-growing field providing a set of new topological and geometric tools to infer relevant features for possibly complex data.
It proposes new well-founded mathematical theories and computational tools that can be used independently or in combination with other data analysis and statistical learning techniques.
This article is a brief introduction, through a few selected topics, to basic fundamental and practical aspects of tda for nonexperts. Topological data analysis tda is a recent field that emerged from various works in applied algebraic topology and computational geometry during the first decade of the century. Although one can trace back geometric approaches to data analysis quite far into the past, tda really started as a field with the pioneering works of Edelsbrunner et al.
Although it is still rapidly evolving, tda now provides a set of mature and efficient tools that can be used in combination with or complementarily to other data science tools. 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, many standard ones rely on the following basic pipeline that will serve as the backbone of this article:.
The input is assumed to be a finite set of points coming with a notion of distanceβor similarityβbetween them. This distance can be induced by the metric in the ambient space e. The definition of the metric on the data is usually given as an input or guided by the application.
