On topological data mining

Abstract

Humans are very good at pattern recognition in dimensions of ≤ 3. However, most of data, e.g. in the biomedical domain, is in dimensions much higher than 3, which makes manual analyses awkward, sometimes practically impossible. Actually, mapping higher dimensional data into lower dimensions is a major task in Human-Computer Interaction and Interactive Data Visualization, and a concerted effort including recent advances in computational topology may contribute to make sense of such data. Topology has its roots in the works of Euler and Gauss, however, for a long time was part of theoretical mathematics. Within the last ten years computational topology rapidly gains much interest amongst computer scientists. Topology is basically the study of abstract shapes and spaces and mappings between them. It originated from the study of geometry and set theory. Topological methods can be applied to data represented by point clouds, that is, finite subsets of the ndimensional Euclidean space. We can think of the input as a sample of some unknown space which one wishes to reconstruct and understand, and we must distinguish between the ambient (embedding) dimension n, and the intrinsic dimension of the data. Whilst n is usually high, the intrinsic dimension, being of primary interest, is typically small. Therefore, knowing the intrinsic dimensionality of data can be seen as one first step towards understanding its structure. Consequently, applying topological techniques to data mining and knowledge discovery is a hot and promising future research area.