Path-connectedness of tensor ranks

Computations of low-rank approximations of tensors often involve path-following optimization algorithms. In such cases, a correct solution may only be found if there exists a continuous path connecting the initial point to a desired solution. We will investigate the existence of such a path in sets of low-rank tensors for various notions of ranks, including tensor rank, border rank, multilinear rank, and their counterparts for symmetric tensors.