2,528 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
Discovering the roots: Uniform closure results for algebraic classes under factoring
Newton iteration (NI) is an almost 350 years old recursive formula that
approximates a simple root of a polynomial quite rapidly. We generalize it to a
matrix recurrence (allRootsNI) that approximates all the roots simultaneously.
In this form, the process yields a better circuit complexity in the case when
the number of roots is small but the multiplicities are exponentially
large. Our method sets up a linear system in unknowns and iteratively
builds the roots as formal power series. For an algebraic circuit
of size we prove that each factor has size at most a
polynomial in: and the degree of the squarefree part of . Consequently,
if is a -hard polynomial then any nonzero multiple
is equally hard for arbitrary positive 's, assuming
that is at most .
It is an old open question whether the class of poly()-sized formulas
(resp. algebraic branching programs) is closed under factoring. We show that
given a polynomial of degree and formula (resp. ABP) size
we can find a similar size formula (resp. ABP) factor in
randomized poly()-time. Consequently, if determinant requires
size formula, then the same can be said about any of its
nonzero multiples.
As part of our proofs, we identify a new property of multivariate polynomial
factorization. We show that under a random linear transformation ,
completely factors via power series roots. Moreover, the
factorization adapts well to circuit complexity analysis. This with allRootsNI
are the techniques that help us make progress towards the old open problems,
supplementing the large body of classical results and concepts in algebraic
circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \&
Burgisser, FOCS 2001).Comment: 33 Pages, No figure
Complete Derandomization of Identity Testing and Reconstruction of Read-Once Formulas
In this paper we study the identity testing problem of arithmetic read-once formulas (ROF) and some related models. A read-once formula is formula (a circuit whose underlying graph is a tree) in which the operations are {+,x} and such that every input variable labels at most one leaf. We obtain the first polynomial-time deterministic identity testing algorithm that operates in the black-box setting for read-once formulas, as well as some other related models. As an application, we obtain the first polynomial-time deterministic reconstruction algorithm for such formulas. Our results are obtained by improving and extending the analysis of the algorithm of [Shpilka-Volkovich, 2015
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
Hitting Sets for Orbits of Circuit Classes and Polynomial Families
The orbit of an n-variate polynomial f(?) over a field ? is the set {f(A?+?) : A ? GL(n,?) and ? ? ??}. In this paper, we initiate the study of explicit hitting sets for the orbits of polynomials computable by several natural and well-studied circuit classes and polynomial families. In particular, we give quasi-polynomial time hitting sets for the orbits of:
1) Low-individual-degree polynomials computable by commutative ROABPs. This implies quasi-polynomial time hitting sets for the orbits of the elementary symmetric polynomials.
2) Multilinear polynomials computable by constant-width ROABPs. This implies a quasi-polynomial time hitting set for the orbits of the family {IMM_{3,d}}_{d ? ?}, which is complete for arithmetic formulas.
3) Polynomials computable by constant-depth, constant-occur formulas. This implies quasi-polynomial time hitting sets for the orbits of multilinear depth-4 circuits with constant top fan-in, and also polynomial-time hitting sets for the orbits of the power symmetric and the sum-product polynomials.
4) Polynomials computable by occur-once formulas
- …