Foliated Geometry of Inverse Problems: Torsion, Curvature Duality, and Near-Associativity

TL;DR

This paper introduces a geometric framework for inverse problems using foliations and dual connections, ensuring unique reconstructions and revealing non-associative structures, with applications in AI and microscopy.

Contribution

It unifies differential geometry with inverse problems, demonstrating how torsion and curvature duality lead to unique solutions and non-associative algebraic structures.

Findings

Vanishing torsion and curvature duality guarantee unique, path-independent reconstruction.

Obstructions lead to non-associative quasigroupoids.

Applications demonstrate practical effectiveness in AI and microscopy.

Abstract

We present a geometric framework for reconstruction problems based on Vaisman foliations and Atiyah--Molino sequences. Independent projections induce transverse foliations and dual connections; vanishing torsion and curvature duality guarantee unique, path-independent reconstruction, while obstructions yield non-associative quasigroupoids. Toric symmetry provides equivariant uniqueness. Applications to generative AI imputation and cryo-electron microscopy demonstrate the framework's practical power, unifying differential geometry with data-driven inverse problems.