Multi-linear formulas for permanent and determinant are of super-polynomial size
Journal of the ACM (JACM)
Tensor-rank and lower bounds for arithmetic formulas
Proceedings of the forty-second ACM symposium on Theory of computing
On the hardness of the noncommutative determinant
Proceedings of the forty-second ACM symposium on Theory of computing
Arithmetic Circuits: A survey of recent results and open questions
Foundations and Trends® in Theoretical Computer Science
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We study a new method for proving lower bounds for subclasses of arithmetic circuits. Roughly speaking, the lower bound is proved by bounding the correlation between the coefficients' vector of a polynomial and the coefficients' vector of any product of two polynomials with disjoint sets of variables. We prove lower bounds for several old and new subclasses of circuits.Monotone Circuits: We prove a tight $2^{\Omega(n)}$ lower bound for the size of monotone arithmetic circuits. The highest previous lower bound was $2^{\Omega(\sqrt{n})}$.Orthogonal Formulas: We prove a tight $2^{\Omega(n)}$ lower bound for the size of orthogonal multilinear formulas (defined, motivated, and studied by Aaronson). Previously, nontrivial lower bounds were only known for subclasses of orthogonal multilinear formulas.Non-Cancelling Formulas: We define and study the new model of {\it non-cancelling multilinear formulas}. Roughly speaking, in this model one is not allowed to sum two polynomials that almost cancel each other. The non-cancelling multilinear model is a generalization of both the monotone model and the orthogonal model. We prove lower bounds of $n^{\Omega(1)}$ for the {\em depth} of non-cancelling multilinear formulas.Noise-Resistant Formulas:We define and study the new model of {\it noise-resistantmultilinear formulas}. Roughly speaking, noise-resistant formulas are formulas that compute the required polynomial even in the presence of noise. We prove lower bounds of $n^{\Omega(1)}$ for the {\em depth} of noise-resistant multilinear formulas.One ingredient of our proof is an explicit map $f:\set{0,1}^n \to \set{0,1}$ that has exponentially small discrepancy for every partition of $\set{1,\ldots,n}$ into two sets of roughly the same size. We give two additional applications of this property to extractors construction and to communication complexity.Mixed-Source Extractors: A mixed-2-source is a source of randomness whose bits arrive from two independent sources (of size $n/2$ each), but they arrive in a fixed but \emph{unknown} order. We are able to extract a linear number of almost perfect random bits from such sources. Communication Complexity: We prove a tight $\Omega(n)$ lower bound for the randomized best-partition communication complexity of $f$. The best lower bound previously known for this model is $\Omega(\sqrt{n})$.