On computing the determinant in small parallel time using a small number of processors
Information Processing Letters
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
Multi-linear formulas for permanent and determinant are of super-polynomial size
STOC '04 Proceedings of the thirty-sixth annual ACM symposium on Theory of computing
Multilinear- NC" " Multilinear- NC"
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Characterizing Valiant's algebraic complexity classes
Journal of Complexity
Lower Bounds for Syntactically Multilinear Algebraic Branching Programs
MFCS '08 Proceedings of the 33rd international symposium on Mathematical Foundations of Computer Science
Balancing Syntactically Multilinear Arithmetic Circuits
Computational Complexity
A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits
SIAM Journal on Computing
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
Arithmetic circuit lower bounds via maxrank
ICALP'13 Proceedings of the 40th international conference on Automata, Languages, and Programming - Volume Part I
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This work deals with the power of linear algebra in the context of multilinear computation. By linear algebra we mean algebraic branching programs (ABPs) which are known to be computationally equivalent to two basic tools in linear algebra: iterated matrix multiplication and the determinant. We compare the computational power of multilinear ABPs to that of multilinear arithmetic formulas, and prove a tight super-polynomial separation between the two models. Specifically, we describe an explicit n-variate polynomial F that is computed by a linear-size multilinear ABP but every multilinear formula computing F must be of size nΩ(log n).