Alignment of vector fields on manifolds via contraction mappings

According to the manifold hypothesis, high-dimensional data can be viewed and meaningfully represented as a lower-dimensional manifold embedded in a higher dimensional feature space. Manifold learning is a part of machine learning where an intrinsic data representation is uncovered based on the manifold hypothesis.
Many manifold learning algorithms were developed. The one called Grassmann & Stiefel eigenmaps (GSE) has been considered in the paper. One of the GSE subproblems is tangent space alignment. The original solution to this problem has been formulated as a generalized eigenvalue problem. In this formulation, it is plagued with numerical instability, resulting in suboptimal solutions to the subproblem and manifold reconstruction problem in general.
We have proposed an iterative algorithm to directly solve the tangent spaces alignment problem. As a result, we have obtained a significant gain in algorithm efficiency and time complexity. We have compared the performance of our method on various model data sets to show that our solution is on par with the approach to vector fields alignment formulated as an optimization on the Stiefel group.