  We show that a generic tensor $T\in \mathbb{F}^{n\times n\times \dots\times
n}$ of order $k$ and CP rank $d$ can be uniquely recovered from $n\log n+dn\log
\log n +o(n\log \log n) $ uniformly random entries with high probability if $d$
and $k$ are constant and $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$. The bound
is tight up to the coefficient of the second leading term and improves on the
existing $O(n^{\frac{k}{2}}{\rm polylog}(n))$ upper bound for order $k$
tensors. The bound is obtained by showing that the projection of the Segre
variety to a random axis-parallel linear subspace preserves $d$-identifiability
with high probability if the dimension of the subspace is $n\log n+dn\log \log
n +o(n\log \log n) $ and $n$ is sufficiently large.
