Shape Theory $\&$ TDA via the Atiyah--Molino Reconstruction

Abstract

Reconstruction problems lie at the very heart of both mathematics and science, posing the fundamental challenge: \emph{How does one reconstruct a hidden structure from incomplete, fragmented, or distorted data?} In this paper, we introduce a new approach that harnesses the insights of the Vaisman Atiyah--Molino framework. In contrast to conventional methods that depend on persistent homology, our approach exploits the concept of the Vaisman centroid---an intrinsic invariant that encapsulates the averaged geometry of a data set---to resolve the inherent ambiguities of inverse problems. In the present paper, we focus on the theory and applications of the Vaisman centroid, offering a new perspective for Topological Data Analysis that eschews persistent homology in favour of a unified geometric paradigm. A subsequent paper will extend these ideas to a complete reconstruction scheme via the Atiyah--Molino framework. Our method provides a robust and computationally tractable framework for the recovery of hidden structures while opening new avenues for the analysis of high-dimensional and noisy data across the mathematical sciences.