Abstract:
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 ``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.