Reconstruction of full rank algebraic branching programs

Kayal, N and Nair, V and Saha, C and Tavenas, S (2017) Reconstruction of full rank algebraic branching programs. In: Leibniz International Proceedings in Informatics, LIPIcs, 79 (1).

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Abstract

An algebraic branching program (ABP) A can be modelled as a product expression X1 · X2 . . .Xd, where X1 and Xd are 1 × w and w × 1 matrices, respectively, and every other Xk 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 Πdk=1Xk. We say A is a full rank ABP if the w2 (d - 2) + 2w linear forms occurring in the matrices X1,X2, . . . ,Xd are F-linearly independent. Our main result is a randomized reconstruction algorithm for full rank ABPs: Given blackbox access to anm-variate polynomial f of degree at mostm, 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 inm and β, where β is the bit length of the coefficients of f . The algorithm works even if Xk is a wk-1 × wk matrix (with w0 = wd = 1), and w = (W1, . . . ,wd-1) is unknown. The result is obtained by designing a randomized polynomial time equivalence test for the family of iterated matrix multiplication polynomial IMMw,d , the (1, 1)-th entry of a product of d rectangular symbolic matrices whose dimensions are according to w ∈ Nd-1. At its core, the algorithm exploits a connection between the irreducible invariant subspaces of the Lie algebra of the group of symmetries of a polynomial f that is equivalent toIMMw,d and the "layer spaces" of a full rank ABP computing f . This connection also helps determine the group of symmetries of IMMw,d and show that IMMw,d is characterized by its group of symmetries.

Item Type:	Journal Article
Publication:	Leibniz International Proceedings in Informatics, LIPIcs
Publisher:	Association for Computing Machinery
Additional Information:	The copyright for this article belongs to the Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing.
Keywords:	Algebra; Computational complexity; Lie groups; Polynomial approximation; Polynomials; Product design, Algebraic branching programs; Circuit reconstruction; Equivalence tests; Lie Algebra; MAtrix multiplication, Matrix algebra
Department/Centre:	Division of Electrical Sciences > Computer Science & Automation
Date Deposited:	17 Jul 2022 06:34
Last Modified:	17 Jul 2022 06:34
URI:	https://eprints.iisc.ac.in/id/eprint/74515

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