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

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.