Reconstruction of Full Rank Algebraic Branching Programs
Résumé
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 X1·X2·...·Xd. We say A is a full rank ABP if the w^2 (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 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 Xk is a w_(k−1)×w_(k) matrix (with w_0 = w_d = 1), and w = (w_1, ..., w_(d−1)) is unknown. The result is obtained by designing a randomized polynomial time equivalence test for the family of iterated matrix multiplication polynomial IMM_(w,d), the (1,1)-th entry of a product of d rectangular symbolic matrices whose dimensions are according to w ∈ N^(d−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 to IMM w,d and the 'layer spaces' of a full rank ABP computing f. This connection also helps determine the group of symmetries of IMM w,d and show that IMM w,d is characterized by its group of symmetries.
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