We show how to extend several basic concentration inequalities for simple
random tensors X=x1βββ―βxdβ where all xkβ are
independent random vectors in Rn with independent coefficients. The
new results have optimal dependence on the dimension n and the degree d. As
an application, we show that random tensors are well conditioned: (1βo(1))nd independent copies of the simple random tensor XβRnd
are far from being linearly dependent with high probability. We prove this fact
for any degree d=o(n/lognβ) and conjecture that it is true for any
d=O(n).Comment: A few more typos were correcte