Reconstruction of Full Rank Algebraic Branching Programs

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 ∏k=1 Xk. 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 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 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 to IMMw,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. ∗Address: CNRS, LAMA, F-73000 Chambéry, France. A part of this work was done during a postdoctoral stay in Microsoft Research India.