Reconstruction of Real Depth-3 Circuits with Top Fan-In 2

We present a polynomial time randomized algorithm for reconstructing $\Sigma\Pi\Sigma(2)$ circuits over $\mathbb{R}$, i.e. depth 3 circuits with fan in 2 at the top addition gate and having real coefficients. The algorithm needs only a blackbox query access to the polynomial $f\in \mathbb{R}[x_1,\ldots,x_n]$ of degree d in n variables, computable by a $\Sigma\Pi\Sigma(2)$ 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 $\Sigma\Pi\Sigma(2)$ circuit(with high probability). Our main techniques are based on the use of Quantitative Syslvester 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 Brills Equations([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].