sheaves

When using sheaves to model real-world problems -- e.g., topological signal processing or information integration -- we must face up to the challenges that real data provide. In a sheaf-theoretical context, measured data, called assignments, are represented by members of objects in the data category. Assignments need not be sections, and indeed, as measured data, almost always require statistical descriptions and tolerances. Uncertainty can exist between disparate measurements generally, and even on the same observable, due to uncertainty, noise, or sensor malfunctions.

A fundamental question in the study of high-dimensional data is as
follows: Given high-dimensional point cloud samples, how can we infer
the structures of the underlying data?
In manifold learning, we assume the data is supported by a
low-dimensional space with a manifold structure.
However, such an assumption may be too restrictive in practice when we are given point cloud samples not of a manifold but of a stratified space, which contain singularities and mixed dimensionality.