An Almost Optimal Rank Bound for Depth-3 Identities

TL;DR

This paper establishes an almost optimal upper bound on the rank of simple, minimal zero depth-3 circuits, significantly improving previous bounds and advancing the understanding of circuit identities and their factorization patterns.

Contribution

It introduces a new concept called 'ideal matchings' to analyze depth-3 circuits, leading to improved rank bounds and structural insights that could enable deterministic identity testing.

Findings

Rank of simple, minimal zero depth-3 circuits is at most k^3 log d.

The rank bound improves previous exponential bounds, reducing dependence on k.

Linear factors of such circuits have rank at most k^3 log d.

Abstract

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 \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 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.