Reconstruction of Depth-4 Multilinear Circuits with Top Fan-in 2

We present a randomized algorithm for reconstructing multilinear Sigma Pi Sigma Pi(2) circuits, i.e., multilinear depth-4 circuits with fan-in 2 at the top + gate. The algorithm is given blackbox access to a polynomial f is an element of F[x(1),..., x(n)] computable by a multilinear Sigma Pi Sigma Pi(2) circuit of size s and outputs an equivalent multilinear Sigma Pi Sigma Pi(2) circuit, runs in time poly(n, s), and works over any field F. This is the first reconstruction result for any model of depth-4 arithmetic circuits. Prior to our work, reconstruction results for bounded depth circuits were known only for depth-2 arithmetic circuits (Klivans & Spielman, STOC 2001), Sigma Pi Sigma(2) circuits (depth-3 arithmetic circuits with top fan-in 2) (Shpilka, STOC 2007), and Sigma Pi Sigma(k) with k = O(1) (Karnin & Shpilka, CCC 2009). Moreover, the running times of these algorithms have a polynomial dependence on vertical bar F vertical bar and hence do not work for infinite fields such as Q. Our techniques are quite different from the previous ones for depth-3 reconstruction and rely on a polynomial operator introduced by Karnin et al. (STOC 2010) and Saraf & Volkovich (STOC 2011) for devising blackbox identity tests for multilinear Sigma Pi Sigma Pi(k) circuits. Some other ingredients of our algorithm include the classical multivariate blackbox factoring algorithm by Kaltofen & Trager (FOCS 1988) and an average-case algorithm for reconstructing Sigma Pi Sigma(2) circuits by Kayal.