**In minisymposium: Low Rank Tensor Approximation**

To avoid non-smooth points in optimizing low-rank tensor approximation problems, it is often desirable to restrict the feasible set to tensors of a specific rank, as opposed to tensors not more than a specific rank. This can potentially create another issue as the set of tensors of a specific rank may well be a disconnected set, and path-following algorithms starting in one component would fail to find an optimizer located in another. This is especially a problem over the reals – for example, it is well-known that the set of real full-rank square matrices is not a connected set. We will discuss path-connectedness of tensor rank, border rank, and multilinear rank. We show that the set of real tensors of a fixed tensor rank or border rank $r$ is always path connected if $r$ is less than the complex generic rank (subgeneric). The set of real $d$-tensors of multilinear rank $(r_1,\dots,r_d)$ has more complicated connectedness properties; we will show that in the subgeneric case it is connected if $r_1 = \dots = r_d$ or if $r_i < \prod_{j \ne i} r_j$ for all $i=1,\dots,d$ but not in general.

This work is supported by ERC AdG-2013-320594 DECODA