ECCC-Report TR16-096https://eccc.weizmann.ac.il/report/2016/096Comments and Revisions published for TR16-096en-usTue, 01 Aug 2017 06:05:49 +0300
Revision 2
| The Chasm at Depth Four, and Tensor Rank : Old results, new insights |
Mrinal Kumar,
Suryajith Chillara,
Ramprasad Saptharishi,
V Vinay
https://eccc.weizmann.ac.il/report/2016/096#revision2Agrawal 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 $\omega(1) \leq d \leq n^{o(1)}$.
Tue, 01 Aug 2017 06:05:49 +0300https://eccc.weizmann.ac.il/report/2016/096#revision2
Revision 1
| The Chasm at Depth Four, and Tensor Rank : Old results, new insights |
Mrinal Kumar,
Suryajith Chillara,
Ramprasad Saptharishi,
V Vinay
https://eccc.weizmann.ac.il/report/2016/096#revision1Agrawal 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 Koiran [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 $\omega(1) \leq d \leq n^{o(1)}$.
Tue, 14 Jun 2016 15:24:06 +0300https://eccc.weizmann.ac.il/report/2016/096#revision1
Paper TR16-096
| The Chasm at Depth Four, and Tensor Rank : Old results, new insights |
Mrinal Kumar,
Suryajith Chillara,
Ramprasad Saptharishi,
V Vinay
https://eccc.weizmann.ac.il/report/2016/096Agrawal 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 $\omega(1) \leq d \leq n^{o(1)}$.
Tue, 14 Jun 2016 12:31:25 +0300https://eccc.weizmann.ac.il/report/2016/096