2008/11/19 by Nitin Saxena, Saxena, Nitin, C. Seshadhri +1
Computer Science · #Computational Complexity (cs.CC) #FOS: Computer and information sciences #cs.CC
paper · pdf · doi:10.48550/arxiv.0811.3161
25 pages, preliminary version
arxiv created 2008/11/19 · arxiv updated 2009/12/01
We show that the rank of a depth-3 circuit (over any field) that is simple, minimal and zero is at most k3log d. The previous best rank bound known was 2O(k2)(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 Ω(klog 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 k3log 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.