604 research outputs found
Reconstruction of Full Rank Algebraic Branching Programs
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 x w and w x 1 matrices respectively, and every other X_k is a w x 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 x 1 matrix obtained from the product X_1 X_2 ... X_d. 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 \u27no full rank ABP exists\u27 (with high probability). The running time of the algorithm is polynomial in m and b, where b is the bit length of the coefficients of f. The algorithm works even if X_k is a w_{k-1} x w_k matrix (with w_0 = w_d = 1), and v = (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_{v,d}, the (1,1)-th entry of a product of d rectangular symbolic matrices whose dimensions are according to v in 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_{v,d} and the \u27layer spaces\u27 of a full rank ABP computing f. This connection also helps determine the group of symmetries of IMM_{v,d} and show that IMM_{v,d} is characterized by its group of symmetries
Randomized Polynomial-Time Equivalence Between Determinant and Trace-IMM Equivalence Tests
Equivalence testing for a polynomial family {g_m} over a field F is the
following problem: Given black-box access to an n-variate polynomial f(x),
where n is the number of variables in g_m, check if there exists an A in
GL(n,F) such that f(x) = g_m(Ax). If yes, then output such an A. The complexity
of equivalence testing has been studied for a number of important polynomial
families, including the determinant (Det) and the two popular variants of the
iterated matrix multiplication polynomial: IMM_{w,d} (the (1,1) entry of the
product of d many w w symbolic matrices) and Tr-IMM_{w,d} (the trace
of the product of d many w w symbolic matrices). The families Det, IMM
and Tr-IMM are VBP-complete, and so, in this sense, they have the same
complexity. But, do they have the same equivalence testing complexity? We show
that the answer is 'yes' for Det and Tr-IMM (modulo the use of randomness). The
result is obtained by connecting the two problems via another well-studied
problem called the full matrix algebra isomorphism problem (FMAI). In
particular, we prove the following:
1. Testing equivalence of polynomials to Tr-IMM_{w,d}, for d 3 and
w 2, is randomized polynomial-time Turing reducible to testing
equivalence of polynomials to Det_w, the determinant of the w w matrix
of formal variables. (Here, d need not be a constant.)
2. FMAI is randomized polynomial-time Turing reducible to equivalence testing
(in fact, to tensor isomorphism testing) for the family of matrix
multiplication tensors {Tr-IMM_{w,3}}.
These in conjunction with the randomized poly-time reduction from determinant
equivalence testing to FMAI [Garg,Gupta,Kayal,Saha19], imply that FMAI,
equivalence testing for Tr-IMM and for Det, and the -tensor isomorphism
problem for the family of matrix multiplication tensors are randomized
poly-time equivalent under Turing reductions.Comment: 36 pages, 2 figure
THE TOOLS AND MONTE CARLO WORKING GROUP Summary Report from the Les Houches 2009 Workshop on TeV Colliders
This is the summary and introduction to the proceedings contributions for the
Les Houches 2009 "Tools and Monte Carlo" working group.Comment: 144 Pages. Workshop site
http://wwwlapp.in2p3.fr/conferences/LesHouches/Houches2009/ . Conveners were
Butterworth, Maltoni, Moortgat, Richardson, Schumann and Skand
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