An Almost Optimal Rank Bound for Depth-3 Identities

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.