📂Topological Data Analysis

Filtration of Complexes

Definition ¹

Let $K$ be a simplicial complex. A subset $L \subset K$ is a Subcomplex of $K$ if it is a simplicial complex itself. $$\emptyset = K^{0} \subset K^{1} \subset \cdots \subset K^{m} = K$$ A Nested Sequence of subcomplexes of $K$ is called the Filtration of $K$. Generally, for all $i \ge m$, it is presumed that $K^{i} = K^{m}$. When such a filtration exists, $K$ is referred to as a Filtered Complex.

Explanation

Ascending Chain

The subcomplexes of the above-filtered complex illustrate a structure similar to an Ascending Chain with respect to $m = 0,\cdots,5$. $$K^{0} \subset K^{1} \subset K^{2} \subset K^{3} \subset K^{4} \subset K^{5}$$ The term filtration suggests an image of the largest simplices being filtered out (in the ←left direction) and gradually becoming smaller, although it is not always necessary in mathematics to imagine it solely in a reducing manner.

Topological Data Analysis

Complexes such as the Vietoris-Rips Complex or the Cech Complex are determined by a given radius $\varepsilon > 0$, and listing the complexes obtained by incrementally increasing this $\varepsilon$ effectively constitutes a filtered complex. Identifying the Persistency of topological properties that appear and disappear within such filtered complexes is a foundational approach of Topological Data Analysis in characterizing the features of data.

See Also

Various Filtrations

$$A_{1} \subset A_{2} \subset \cdots \subset A_{n} \subset \cdots$$ Commonly, in mathematics, when a structure forms a Nested Sequence, it is referred to as a Filtration.

• Filtration of a Stochastic Process
• Filtration of Complexes
• Flag of Vector Spaces