  Agrawal and Vinay [AV08] showed how any polynomial size arithmetic circuit
can be thought of as a depth four arithmetic circuit of subexponential size.
The resulting circuit size in this simulation was more carefully analyzed by
Korian [Koiran] and subsequently by Tavenas [Tav13]. We provide a simple proof
of this chain of results. We then abstract the main ingredient to apply it to
formulas and constant depth circuits, and show more structured depth reductions
for them.
  In an apriori surprising result, Raz [Raz10] showed that for any $n$ and $d$,
such that $ \omega(1) \leq d \leq O\left(\frac{\log n}{\log\log n}\right)$,
constructing explicit tensors $T:[n]^d \rightarrow F$ of high enough rank would
imply superpolynomial lower bounds for arithmetic formulas over the field $F$.
Using the additional structure we obtain from our proof of the depth reduction
for arithmetic formulas, we give a new and arguably simpler proof of this
connection. We also extend this result for homogeneous formulas to show that,
in fact, the connection holds for any $d$ such that $\omega(1) \leq d \leq
n^{o(1)}$.
