What is 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 invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. (Wikpedia)
What is Big Data:
Big data is a broad term for data sets so large or complex that traditional data processing applications are inadequate. Challenges include analysis, capture, data curation, search, sharing, storage, transfer, visualization, and information privacy. The term often refers simply to the use of predictive analytics or other certain advanced methods to extract value from data, and seldom to a particular size of data set. Accuracy in big data may lead to more confident decision making. And better decisions can mean greater operational efficiency, cost reductions and reduced risk. (Wikipedia)
Big Data is said to be the next biggest scientific advance since the internet. Algebraic Topology is one branch of Mathematics that is directly related to Big Data.
- 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, e.g. 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 persistence diagram or barcode.