article

Random arithmetic formulas can be reconstructed efficiently

Authors:

Youming QiaoAuthors Info & Claims

Computational Complexity, Volume 23, Issue 2

Pages 207 - 303

https://doi.org/10.1007/s00037-014-0085-0

Published: 01 June 2014 Publication History

Abstract

Informally stated, we present here a randomized algorithm that given black-box access to the polynomial f computed by an unknown/hidden arithmetic formula reconstructs, on the average , an equivalent or smaller formula $${\hat{\phi}}$$ ^ in time polynomial in the size of its output $${\hat{\phi}}$$ ^ .

Specifically, we consider arithmetic formulas wherein the underlying tree is a complete binary tree, the leaf nodes are labeled by affine forms (i.e., degree one polynomials) over the input variables and where the internal nodes consist of alternating layers of addition and multiplication gates. We call these alternating normal form (ANF) formulas. If a polynomial f can be computed by an arithmetic formula μ of size s , it can also be computed by an ANF formula , possibly of slightly larger size s ^{O (1)}. Our algorithm gets as input black-box access to the output polynomial f (i.e., for any point x in the domain, it can query the black box and obtain f ( x ) in one step) of a random ANF formula of size s (wherein the coefficients of the affine forms in the leaf nodes of are chosen independently and uniformly at random from a large enough subset of the underlying field). With high probability (over the choice of coefficients in the leaf nodes), the algorithm efficiently (i.e., in time s ^{O (1)}) computes an ANF formula $${\hat{\phi}}$$ ^ of size s computing f . This then is the strongest model of arithmetic computation for which a reconstruction algorithm is presently known, albeit efficient in a distributional sense rather than in the worst case.

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Cited By

Medini DShpilka A(2021)Hitting sets and reconstruction for dense orbits in VPe and ΣΠΣ circuitsProceedings of the 36th Computational Complexity Conference10.4230/LIPIcs.CCC.2021.19Online publication date: 20-Jul-2021
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2021.19
Bhargava VSaraf SVolkovich IKhuller SVassilevska Williams V(2021)Reconstruction algorithms for low-rank tensors and depth-3 multilinear circuitsProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451096(809-822)Online publication date: 15-Jun-2021
https://dl.acm.org/doi/10.1145/3406325.3451096
Mukhopadhyay PQiao Y(2017)Sparse multivariate polynomial interpolation on the basis of Schubert polynomialsComputational Complexity10.1007/s00037-016-0142-y26:4(881-909)Online publication date: 1-Dec-2017
https://dl.acm.org/doi/10.1007/s00037-016-0142-y
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Random arithmetic formulas can be reconstructed efficiently
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cover image Computational Complexity

Computational Complexity Volume 23, Issue 2

June 2014

204 pages

ISSN:1016-3328

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Copyright © Copyright © 2014 Springer Basel.

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Birkhauser Verlag

Switzerland

Publication History

Published: 01 June 2014

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Cited By

Medini DShpilka A(2021)Hitting sets and reconstruction for dense orbits in VPe and ΣΠΣ circuitsProceedings of the 36th Computational Complexity Conference10.4230/LIPIcs.CCC.2021.19Online publication date: 20-Jul-2021
https://dl.acm.org/doi/10.4230/LIPIcs.CCC.2021.19
Bhargava VSaraf SVolkovich IKhuller SVassilevska Williams V(2021)Reconstruction algorithms for low-rank tensors and depth-3 multilinear circuitsProceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing10.1145/3406325.3451096(809-822)Online publication date: 15-Jun-2021
https://dl.acm.org/doi/10.1145/3406325.3451096
Mukhopadhyay PQiao Y(2017)Sparse multivariate polynomial interpolation on the basis of Schubert polynomialsComputational Complexity10.1007/s00037-016-0142-y26:4(881-909)Online publication date: 1-Dec-2017
https://dl.acm.org/doi/10.1007/s00037-016-0142-y
Sinha G(2016)Reconstruction of real depth-3 circuits with top fan-in 2Proceedings of the 31st Conference on Computational Complexity10.5555/2982445.2982476(1-53)Online publication date: 29-May-2016
https://dl.acm.org/doi/10.5555/2982445.2982476

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