@inproceedings{gupta2012reconstruction,
author = {Gupta, Ankit and Kayal, Neeraj and Lokam, Satya},
title = {Reconstruction of Depth-4 Multilinear Circuits with Top Fan-in Two},
booktitle = {Symposium on Theory of Computing (STOC)},
year = {2012},
month = {January},
abstract = {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[x1,..,xn] 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.},
publisher = {ACM},
url = {http://approjects.co.za/?big=en-us/research/publication/reconstruction-of-depth-4-multilinear-circuits-with-top-fanin-two/},
edition = {Symposium on Theory of Computing (STOC)},
}