  In this paper we propose new techniques to sample arbitrary third-order
tensors, with an objective of speeding up tensor algorithms that have recently
gained popularity in machine learning. Our main contribution is a new way to
select, in a biased random way, only $O(n^{1.5}/\epsilon^2)$ of the possible
$n^3$ elements while still achieving each of the three goals: \\ {\em (a)
tensor sparsification}: for a tensor that has to be formed from arbitrary
samples, compute very few elements to get a good spectral approximation, and
for arbitrary orthogonal tensors {\em (b) tensor completion:} recover an
exactly low-rank tensor from a small number of samples via alternating least
squares, or {\em (c) tensor factorization:} approximating factors of a low-rank
tensor corrupted by noise. \\ Our sampling can be used along with existing
tensor-based algorithms to speed them up, removing the computational bottleneck
in these methods.
