Topological data analysis
Topological data analysis (TDA) is a relatively new area of research that spans many disciplines including topology (in particular, homology), statistics, machine learning and computation geometry.
From Wikipedia:

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 theory of persistent homology.[1]
  3. Encode the persistent homology of a data set in the form of a parameterized version of a Betti number which is called a barcode.[2]
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Topological data analysis
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