We present a randomized algorithm for reconstructing multilinear depth-4
arithmetic circuits with fan-in 2 at the top + gate. The algorithm is given
blackbox access to a multilinear polynomial f in
F[x_{1},..,x_{n}] computable by a multilinear
Sum-Product-Sum-Product(SPSP) circuit of size s and outputs an equivalent
multilinear SPSP circuit, runs in time poly(ns) 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), SPS circuits (depth-3 arithmetic circuits with top fan-in 2) (Shpilka, STOC 2007), and SPS(k) with k=O(1) (Karnin & Shpilka, CCC 2009). Moreover, the running times of these algorithms have a polynomial dependence on |F| 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 SPSP(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 SPS circuits by Kayal.

}, author = {Ankit Gupta and Neeraj Kayal and Satya Lokam}, booktitle = {Symposium on Theory of Computing (STOC)}, publisher = {ACM}, title = {Reconstruction of Depth-4 Multilinear Circuits with top fanin two}, url = {http://research.microsoft.com/apps/pubs/default.aspx?id=158918}, year = {2012}, }