{"id":162374,"date":"2012-01-01T00:00:00","date_gmt":"2012-01-01T00:00:00","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/reconstruction-of-depth-4-multilinear-circuits-with-top-fanin-two\/"},"modified":"2018-10-16T20:16:55","modified_gmt":"2018-10-17T03:16:55","slug":"reconstruction-of-depth-4-multilinear-circuits-with-top-fanin-two","status":"publish","type":"msr-research-item","link":"https:\/\/www.microsoft.com\/en-us\/research\/publication\/reconstruction-of-depth-4-multilinear-circuits-with-top-fanin-two\/","title":{"rendered":"Reconstruction of Depth-4 Multilinear Circuits with Top Fan-in Two"},"content":{"rendered":"<div class=\"asset-content\">\n<p>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<sub>1<\/sub>,..,x<sub>n<\/sub>] 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.<\/p>\n<p>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.<\/p>\n<p>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.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>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 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