Topological data analysis (TDA) is a new area of study aimed at having applications in areas such as data mining and computer vision. The main problems are:
- how one infers high-dimensional structure from low-dimensional representations; and
- 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:
- Replace a set of data points with a family of simplicial complexes, indexed by a proximity parameter.
- Analyse these topological complexes via algebraic topology — specifically, via the theory of persistent homology.
- Encode the persistent homology of a data set in the form of a parameterized version of a Betti number which is called a barcode.
Read more about Topological Data Analysis: Point Cloud Data, Background, Combinatorial Representations, Topological Invariants, Multiscale Invariants, See Also
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