Leveraging the Power of Signatures for the Construction of Topological Complexes for the Analysis of Multivariate Complex Dynamics

Topological Data Analysis is a field of great interest in many applications such as finance or neuroscience. The goal of the present paper is to propose a novel approach to building simplicial complexes that capture the multiway ordered interactions in the components of high-dimensional time series using the theory of Signatures. Signatures represent one of the most powerful transforms for extracting group-wise structural features and we put them to work in the task of discovering statistically meaningful simplices from a complex that we estimate sequentially. Numerical experiments on an fMRI dataset illustrates the efficiency and relevance of our approach.