Topological data analysis

Topological data analysis is a new area of study aimed at having applications in areas such as data mining and computer vision. The main problems are (1) how one infers high-dimensional structure from low-dimensional representations; and (2) how one assembles discrete points into global structure.

The human brain can easily extract global structure from representations in a strictly lower dimension, i.e. we infer a 3D environment from a 2D image from each eye. The inference of global structure also occurs when converting discrete data into continuous images. E.g. dot-matrix printers and televisions communicate images via arrays of discrete points.

The main method used by topological data analysis is:

(1) replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter.

(2) Analyse these topological complexes via algebraic topology — specifically, via the new theory of persistent homology.

(3) Encode the persistent homology of a data set in the form of a parameterized version of a Betti number which will be called a barcode.

Point cloud data
Data is often represented as points in a Euclidean n-dimensional space En. The global shape of the data may provide information about the phenomena that the data represent.

One type of data set for which global features are certainly present is the so-called point cloud data coming from physical objects in 3D. E.g. a laser can scan an object at a set of discrete points and the cloud of such points can be used in a computer representation of the object. Point cloud data refers to any collection of points in En or a (perhaps noisy) sample of points on a lower-dimensional subset.

For point clouds in low-dimensional spaces there are numerous approaches for inferring features based on planar projections in the fields of computer graphics and statistics. Topological data analysis is needed when the spaces are high-dimensional or too twisted to allow planar projections.

To convert a point cloud in a metric space into a global object use the point cloud as the vertices of a graph whose edges are determined by proximity, then turn the graph into a simplicial complex and use algebraic topology to study it.