Reconstruction of real depth-3 circuits with top fan-in 2

Abstract

Reconstruction of arithmetic circuits has been heavily studied in the past few years and has connections to proving lower bounds and deterministic identity testing. In this paper we present a polynomial time randomized algorithm for reconstructing ΣΠΣ(2) circuits over F (char(F) = 0), i.e. depth-3 circuits with fan-in 2 at the top addition gate and having coefficients from a field of characteristic 0. The algorithm needs only a blackbox query access to the polynomial f ∈ F[x1, ..., xn] of degree d, computable by a ΣΠΣ(2) circuit C. In addition, we assume that the simple rank of this polynomial (essential number of variables after removing the gcd of the two multiplication gates) is bigger than a fixed constant. Our algorithm runs in time poly(n, d) and returns an equivalent ΣΠΣ(2) circuit (with high probability). The problem of reconstructing ΣΠΣ(2) circuits over finite fields was first proposed by Shpilka [24]. The generalization to ΣΠΣ(k) circuits, k = O(1) (over finite fields) was addressed by Karnin and Shpilka in [15]. The techniques in these previous involve iterating over all objects of certain kinds over the ambient field and thus the running time depends on the size of the field F. Their reconstruction algorithm uses lower bounds on the lengths of Linear Locally Decodable Codes with 2 queries. In our settings, such ideas immediately pose a problem and we need new ideas to handle the case of the characteristic 0 field F. Our main techniques are based on the use of Quantitative Sylvester Gallai Theorems from the work of Barak et al. [3] to find a small collection of nice subspaces to project onto. The heart of our paper lies in subtle applications of the Quantitative Sylvester Gallai theorems to prove why projections w.r.t. the nice subspaces can be glued. We also use Brill's Equations from [8] to construct a small set of candidate linear forms (containing linear forms from both gates). Another important technique which comes very handy is the polynomial time randomized algorithm for factoring multivariate polynomials given by Kaltofen [14].