Topological Data Analysis For Pattern Discovery In High Dimensional Mathematical Spaces

Authors: Dr A.Srikrishna Chaitanya, Dr P.Srilakshmi

Abstract: Mathematical spaces of high dimension are often characterized by nonlinear, nonconvex, and multiscale properties, which cannot be adequately analyzed by the classical distance-based and/or variance approaches. Topological data analysis (TDA) is a remedy to this problem by means of finding scale-invariant topological descriptors, such as connected components, cycles, voids, etc. This work proposes a method for discovering patterns in high-dimensional mathematical spaces using the techniques of persistent homology and the Mapper algorithm in combination with vectorization of topological descriptors. This approach is tested on synthetic point clouds drawn from known manifolds (torus, sphere, and clustered subspaces) and high-dimensional algebraic varieties samples. The comparison with the classical methods of principal component analysis and k-means clustering reveals that topological descriptors are able to detect structure information that cannot be detected by the linear projection approaches.

DOI: https://doi.org/10.5281/zenodo.23906854