  We develop the first fast spectral algorithm to decompose a random
third-order tensor over $\mathbb{R}^d$ of rank up to
$O(d^{3/2}/\text{polylog}(d))$. Our algorithm only involves simple linear
algebra operations and can recover all components in time $O(d^{6.05})$ under
the current matrix multiplication time.
  Prior to this work, comparable guarantees could only be achieved via
sum-of-squares [Ma, Shi, Steurer 2016]. In contrast, fast algorithms [Hopkins,
Schramm, Shi, Steurer 2016] could only decompose tensors of rank at most
$O(d^{4/3}/\text{polylog}(d))$.
  Our algorithmic result rests on two key ingredients. A clean lifting of the
third-order tensor to a sixth-order tensor, which can be expressed in the
language of tensor networks. A careful decomposition of the tensor network into
a sequence of rectangular matrix multiplications, which allows us to have a
fast implementation of the algorithm.
