A homogeneous geometry of low-rank tensors

Abstract

We consider sets of fixed CP, multilinear, and TT rank tensors, and derive conditions for when (the smooth parts of) these sets are smooth homogeneous manifolds. For CP and TT ranks, the conditions are essentially that the rank is sufficiently low. These homogeneous structures are then used to derive Riemannian metrics whose geodesics are both complete and efficient to compute.