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2009 24th Annual IEEE Conference on Computational Complexity (2009)
Paris, France
July 15, 2009 to July 18, 2009
ISBN: 978-0-7695-3717-7
pp: 137-148
We show that the rank of a depth-$3$ circuit (over any field) that is simple, minimal and zero is at most $O(k^3\log d)$. The previous best rank bound known was $2^{O(k^2)}(\log d)^{k-2}$ by Dvir and Shpilka (STOC 2005). This almost resolves the rank question first posed by Dvir and Shpilka (as we also provide a simple and minimal identity of rank $\Omega(k\log d)$). Our rank bound significantly improves (dependence on $k$ exponentially reduced) the best known deterministic black-box identity tests for depth-$3$ circuits by Karnin and Shpilka (CCC 2008). Our techniques also shed light on the factorization pattern of nonzero depth-$3$ circuits, most strikingly: the rank of linear factors of a simple, minimal and nonzero depth-$3$ circuit (over any field) is at most $O(k^3\log d)$. The novel feature of this work is a new notion of maps between sets of linear forms, called \emph{ideal matchings}, used to study depth-$3$ circuits. We prove interesting structural results about depth-$3$ identities using these techniques. We believe that these can lead to the goal of a deterministic polynomial time identity test for these circuits.
Identity testing, Derandomization, Depth-3 circuits

C. Seshadhri and N. Saxena, "An Almost Optimal Rank Bound for Depth-3 Identities," 2009 24th Annual IEEE Conference on Computational Complexity(CCC), Paris, France, 2009, pp. 137-148.
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