An Almost Optimal Rank Bound for Depth-3 Identities
| dc.creator | Saxena, Nitin | |
| dc.creator | Seshadhri, C. | |
| dc.date | 2008-11-19 | |
| dc.date.accessioned | 2026-07-07T10:19:31Z | |
| dc.date.available | 2026-07-07T10:19:31Z | |
| dc.description | We show that the rank of a depth-3 circuit (over any field) that is simple, minimal and zero is at most 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 Ω(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 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. | |
| dc.description | 25 pages, preliminary version | |
| dc.identifier | https://arxiv.org/abs/0811.3161 | |
| dc.identifier | http://arxiv.org/abs/0811.3161 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/174539 | |
| dc.subject | Computational Complexity | |
| dc.title | An Almost Optimal Rank Bound for Depth-3 Identities | |
| dc.type | text |