Efficient Reconstruction of Random Multilinear Formulas

Gupta, Ankit; Kayal, Neeraj; Lokam, Satya

doi:10.1109/focs.2011.70

Cited by 10 publications

(8 citation statements)

References 11 publications

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“…Let f ∈ F[Xn] be the polynomial computed by a multilinear ΣΠΣΠ(2) circuit C of size s with ||C|| ≥ 2s 3 as given by equation (10). Then, for any s-sparse multilinear polynomial R, R | f ⇐⇒ R | G.…”

Section: Obtaining Blackbox Access To Sim(c)mentioning

confidence: 99%

“…We nondeterministically guess the 4-tuple of variables (x, y, u, v) satisfying the above properties 10 . So given f , how do we determine the Pi's and Qi's (upto scalar multiples)?…”

Section: Basic Idea and Approachmentioning

confidence: 99%

“…Very recently, in [10] we were able to make progress in this direction for "random" multilinear formulas i.e. formulas sampled from a certain natural distribution over the set of multilinear formulas.…”

mentioning

confidence: 99%

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Reconstruction of depth-4 multilinear circuits with top fan-in 2

Gupta

Kayal

Lokam

2012

Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing

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We present a randomized algorithm for reconstructing multilinear ΣΠΣΠ(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 ∈ F[x1, . . . , xn] computable by a multilinear ΣΠΣΠ(2) circuit of size s and outputs an equivalent multilinear ΣΠΣΠ(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), ΣΠΣ(2) circuits (depth-3 arithmetic circuits with top fan-in 2) (Shpilka, STOC 2007), and ΣΠΣ(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 ΣΠΣΠ(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 ΣΠΣ(2) circuits by Kayal.

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Section: Obtaining Blackbox Access To Sim(c)mentioning

confidence: 99%

“…We nondeterministically guess the 4-tuple of variables (x, y, u, v) satisfying the above properties 10 . So given f , how do we determine the Pi's and Qi's (upto scalar multiples)?…”

Section: Basic Idea and Approachmentioning

confidence: 99%

See 1 more Smart Citation

Reconstruction of depth-4 multilinear circuits with top fan-in 2

Gupta

Kayal

Lokam

2012

Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing

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“…Another way to moderate the reconstruction setup is given by average-case reconstruction. Here the input polynomial f is picked according to some 'natural' distribution on circuits from a class C. This relaxation led to the development of randomized polynomial time reconstruction algorithm for some powerful circuit classes [17,19] (albeit on average), including arithmetic formulas for which we do not know of any super-polynomial lower bound. The notion of average-case reconstruction is related to pseudo-random polynomial Average-case ABP reconstruction: In order to study average-case complexity of the reconstruction problem for ABPs, we need to define a distribution on polynomials computed by ABPs.…”

Section: Circuit Reconstructionmentioning

confidence: 99%

“…There is a randomized algorithm that takes as input a blackbox for an m variate polynomial f over F of degree d ∈ [5, m], and with high probability it does the following: if f is computed by a full rank ABP then the algorithm outputs a full rank ABP computing f , else it outputs 'f does not admit a full rank ABP'. The running time is poly(m, β) 17 , where β is the bit length of the coefficients of f . Remarks: Theorem 5 implies an efficient average-case reconstruction algorithm for ABPs (Problem 3) when m ≥ w 2 d and |S γ | ≥ (mwd) 2 , as a random (w, d, m, S γ )-ABP is full rank with high probability if m and |S γ | are sufficiently large.…”

Section: Definition 4 (Full Rank Algebraic Branching Program)mentioning

confidence: 99%

Reconstruction of Full Rank Algebraic Branching Programs

Kayal

Nair

Saha

et al. 2018

ACM Trans. Comput. Theory

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An algebraic branching program (ABP) A can be modelled as a product expression X 1 • X 2. .. X d , where X 1 and X d are 1×w and w ×1 matrices respectively, and every other X k is a w ×w matrix; the entries of these matrices are linear forms in m variables over a field F (which we assume to be either Q or a field of characteristic poly(m)). The polynomial computed by A is the entry of the 1 × 1 matrix obtained from the product d k=1 X k. We say A is a full rank ABP if the w 2 (d − 2) + 2w linear forms occurring in the matrices X 1 , X 2 ,. .. , X d are F-linearly independent. Our main result is a randomized reconstruction algorithm for full rank ABPs: Given blackbox access to an m-variate polynomial f of degree at most m, the algorithm outputs a full rank ABP computing f if such an ABP exists, or outputs 'no full rank ABP exists' (with high probability). The running time of the algorithm is polynomial in m and β, where β is the bit length of the coefficients of f. The algorithm works even if X k is a w k−1 × w k matrix (with w 0 = w d = 1), and w = (w 1 ,. .. , w d−1) is unknown.

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