Learning barycenters from signature tensors

The expected signature of a family of paths need not be a signature of a path itself. Motivated by this, we consider the notion of a Lie group barycenter introduced by Buser and Karcher to propose a barycenter on signature tensors. We investigate affine algebraic varieties arising from barycenters of several families of samples in paths space, and use path learning techniques (Pfeffer, Seigal, Sturmfels) to recover the underlying path associated to the Lie group barycenter. We provide an implementation in the computer algebra system OSCAR. This is joint work with Carlos Amendola.