Efficient reconstruction of depth three circuits with top fan-in two

03/12/2021

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by   Gaurav Sinha, et al.

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We develop efficient randomized algorithms to solve the black-box reconstruction problem for polynomials over finite fields, computable by depth three arithmetic circuits with alternating addition/multiplication gates, such that output gate is an addition gate with in-degree two. These circuits compute polynomials of form G×(T_1 + T_2), where G,T_1,T_2 are product of affine forms, and polynomials T_1,T_2 have no common factors. Rank of such a circuit is defined as dimension of vector space spanned by all affine factors of T_1 and T_2. For any polynomial f computable by such a circuit, rank(f) is defined to be the minimum rank of any such circuit computing it. Our work develops randomized reconstruction algorithms which take as input black-box access to a polynomial f (over finite field 𝔽), computable by such a circuit. Here are the results. 1 [Low rank]: When 5≤ rank(f) = O(log^3 d), it runs in time (nd^log^3dlog |𝔽|)^O(1), and, with high probability, outputs a depth three circuit computing f, with top addition gate having in-degree ≤ d^rank(f). 2 [High rank]: When rank(f) = Ω(log^3 d), it runs in time (ndlog |𝔽|)^O(1), and, with high probability, outputs a depth three circuit computing f, with top addition gate having in-degree two. Ours is the first blackbox reconstruction algorithm for this circuit class, that runs in time polynomial in log |𝔽|. This problem has been mentioned as an open problem in [GKL12] (STOC 2012)