  We prove that any unconditional set in $\mathbb{R}^N$ that is invariant under
cyclic shifts of coordinates is rigid in $\ell_q^N$, $1\le q\le 2$, i.e. it can
not be well approximated by linear spaces of dimension essentially smaller than
$N$. We apply the approach of E.D.~Gluskin to the setting of averaged
Kolmogorov widths of unconditional random vectors or vectors of independent
mean zero random variables, and prove their rigidity. These results are
obtained using a general lower bound for the averaged Kolmogorov width via weak
moments of biorthogonal random vector. This paper continues the study of the
rigidity initiated by the first author.
  We also provide several corollaries including lower bounds for Kolmogorov
widths of mixed norm balls.
