There is a natural invariant metric on the space of tensors, called Frobenius or Bombieri-Weyl metric. In optimization setting one considers the (complex) critical points on the Segre variety of the distance function from a given tensor, they were characterized by Lim and recently reviewed in Seigal's IPAM tutorial. They are called singular t-ples, among them there is the best rank one approximation.
The geometry of the critical points is appealing, since they lie in a linear space called critical space. We expose some properties of singular t-ples in connection with the critical space.
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