algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

and

topological data analysis In applied mathematics, topological data analysis (TDA) is an approach to the analysis of datasets using techniques from topology. Extraction of information from datasets that are high-dimensional, incomplete and noisy is generally challenging. TDA ...

, the Čech complex is an

abstract simplicial complex In combinatorics, an abstract simplicial complex (ASC), often called an abstract complex or just a complex, is a family of sets that is closed under taking subsets, i.e., every subset of a set in the family is also in the family. It is a purely c ...

constructed from a point cloud in any

metric space In mathematics, a metric space is a Set (mathematics), set together with a notion of ''distance'' between its Element (mathematics), elements, usually called point (geometry), points. The distance is measured by a function (mathematics), functi ...

which is meant to capture

topological Topology (from the Greek words , and ) is the branch of mathematics concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, wit ...

information about the point cloud or the distribution it is drawn from. Given a finite point cloud ''X'' and an ''ε'' > 0, we construct the Čech complex

\check C_\varepsilon(X)

as follows: Take the elements of ''X'' as the vertex set of

\check C_\varepsilon(X)

. Then, for each

\sigma\subset X

, let

\sigma\in \check C_\varepsilon(X)

if the set of ''ε''-balls centered at points of σ has a

nonempty In mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, whi ...

intersection In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their ...

. In other words, the Čech complex is the

nerve A nerve is an enclosed, cable-like bundle of nerve fibers (called axons). Nerves have historically been considered the basic units of the peripheral nervous system. A nerve provides a common pathway for the Electrochemistry, electrochemical nerv ...

of the set of ''ε''-balls centered at points of ''X''. By the nerve lemma, the Čech complex is homotopy equivalent to the union of the balls, also known as the offset filtration.

Relation to Vietoris–Rips complex

The Čech complex is a subcomplex of the

Vietoris–Rips complex In topology, the Vietoris–Rips complex, also called the Vietoris complex or Rips complex, is a way of forming a topological space from distances in a set of points. It is an abstract simplicial complex that can be defined from any metric space ...

. While the Čech complex is more computationally expensive than the Vietoris–Rips complex, since we must check for higher order intersections of the balls in the complex, the nerve theorem provides a guarantee that the Čech complex is homotopy equivalent to union of the balls in the complex. The Vietoris–Rips complex may not be.

References

{{reflist Algebraic topology Computational topology

Relation to Vietoris–Rips complex

See also

References