Annual review of computer science: vol. 3, 1988
Journal of Algorithms
Lower bounds for non-commutative computation
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
An exponential lower bound for depth 3 arithmetic circuits
STOC '98 Proceedings of the thirtieth annual ACM symposium on Theory of computing
Lower bounds on arithmetic circuits via partial derivatives
Computational Complexity
Completeness classes in algebra
STOC '79 Proceedings of the eleventh annual ACM symposium on Theory of computing
Multilinear formulas and skepticism of quantum computing
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Arithmetic Circuits: A Chasm at Depth Four
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Multi-linear formulas for permanent and determinant are of super-polynomial size
Journal of the ACM (JACM)
A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits
SIAM Journal on Computing
Multilinear formulas, maximal-partition discrepancy and mixed-sources extractors
Journal of Computer and System Sciences
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
Tensor Rank: Some Lower and Upper Bounds
CCC '11 Proceedings of the 2011 IEEE 26th Annual Conference on Computational Complexity
Homogeneous Formulas and Symmetric Polynomials
Computational Complexity
Hi-index | 0.00 |
We show that any explicit example for a tensor A : [n]r → F with tensor-rank ≥ nrċ(1−o(1)), where r = r(n) ≤ log n/log log n is super-constant, implies an explicit super-polynomial lower bound for the size of general arithmetic formulas over F. This shows that strong enough lower bounds for the size of arithmetic formulas of depth 3 imply super-polynomial lower bounds for the size of general arithmetic formulas. One component of our proof is a new approach for homogenization and multilinearization of arithmetic formulas, that gives the following results: We show that for any n-variate homogeneous polynomial f of degree r, if there exists a (fanin-2) formula of size s and depth d for f then there exists a homogeneous formula of size O((d+r+1 r) ċ s) for f. In particular, for any r ≤ O(log n), if there exists a polynomial size formula for f then there exists a polynomial size homogeneous formula for f. This refutes a conjecture of Nisan and Wigderson [1996] and shows that super-polynomial lower bounds for homogeneous formulas for polynomials of small degree imply super-polynomial lower bounds for general formulas. We show that for any n-variate set-multilinear polynomial f of degree r, if there exists a (fanin-2) formula of size s and depth d for f, then there exists a set-multilinear formula of size O((d + 2)r ċ s) for f. In particular, for any r ≤ O(log n/log log n), if there exists a polynomial size formula for f then there exists a polynomial size set-multilinear formula for f. This shows that super-polynomial lower bounds for set-multilinear formulas for polynomials of small degree imply super-polynomial lower bounds for general formulas.