In this talk, we introduce a new method for manifold learning,
generalising the LLE (Locally Linear Embedding) algorithm to
manifold-valued data. LLE mostly consists in describing the data with
barycentric coordinates. Since these can't be written in closed form in
a non-Euclidean framework, the generalisation leads to a non-trivial
minimisation problem. We propose a solution which strongly relies on the
computation of the parallel transport of the manifold in a
differentiable way. We implemented the method for the specific case of
Kendall shape spaces, for which we have computations of the parallel
transport compatible with automatic differentiation. We hope for it to
extend to more general manifolds apart from Kendall shape spaces.
Another point is to understand further the properties of the embedding,
especially how they interact with the data. We would expect our method
to work better for data in low concentration or non-homogeneously
distributed, such that the local linear assumption on which the
classical LLE algorithm is based does not hold anymore.