Topology of tensor ranks
Pierre Comon, Lek-Heng Lim, Yang Qi, and Ke Ye

TL;DR
This paper investigates the topological properties, including path-connectedness and homotopy groups, of sets of tensors characterized by various rank notions over real and complex fields, revealing conditions for connectedness and computing homotopy groups.
Contribution
It provides new results on the topological structure of tensor rank sets, including path-connectedness criteria and homotopy group computations over real and complex numbers.
Findings
Set of rank-$r$ tensors is path-connected over $ ext{C}$ if $r$ is not more than the generic rank.
Over $ ext{R}$, rank-$r$ tensors are path-connected if of expected dimension; symmetric cases depend on tensor order.
Homotopy groups of tensor rank sets are computed, revealing topological complexity and differences between real and complex cases.
Abstract
We study path-connectedness and homotopy groups of sets of tensors defined by tensor rank, border rank, multilinear rank, as well as their symmetric counterparts for symmetric tensors. We show that over , the set of rank- tensors and the set of symmetric rank- symmetric tensors are both path-connected if is not more than the complex generic rank; these results also extend to border rank and symmetric border rank over . Over , the set of rank- tensors is path-connected if it has the expected dimension but the corresponding result for symmetric rank- symmetric -tensors depends on the order : connected when is odd but not when is even. Border rank and symmetric border rank over have essentially the same path-connectedness properties as rank and symmetric rank over . When is greater than the…
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