# Arithmetic Circuit Lower Bounds via MaxRank

@article{Kumar2013ArithmeticCL,
title={Arithmetic Circuit Lower Bounds via MaxRank},
author={Mrinal Kumar and Gaurav Maheshwari and Jayalal Sarma},
journal={ArXiv},
year={2013},
volume={abs/1302.3308}
}
• Published 13 February 2013
• Computer Science, Mathematics
• ArXiv
We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits. In particular, we obtain the following results : · As our first main result, we prove that any homogeneous depth-3 circuit for computing the product of d matrices of dimension n ×n requires Ω(nd−1/2d…
Lower Bounds for Projections of Power Symmetric Polynomials
• Mathematics, Computer Science
• 2016
Lower bounds imply deterministic npoly(logn) black-box identity testing algorithms for the above classes of arithmetic circuits.
Limitations of sum of products of Read-Once Polynomials
• Mathematics, Computer Science
Electron. Colloquium Comput. Complex.
• 2015
A class of formulas of unbounded depth with exponential size lower bound against the permanent can be seen as an exponential improvement over the multilinear formula size lower bounds given by Raz for a sub-class of multi-linear and non-multi-linear formulas.
On Constant Depth Circuits Parameterized by Degree: Identity Testing and Depth Reduction
• Mathematics
COCOON
• 2017
The notion of fixed parameter tractability is defined and it is shown that there are families of polynomials of degree k that cannot be computed by homogeneous depth four $$\varSigma \varPi ^{\sqrt{k}}\varS Sigma \var Pi ^{k}$$ circuits, implying that there is no parameterized depth reduction for circuits of size f(k)n^{O(1) such that the resulting depth four circuit is homogeneous.
Towards an algebraic natural proofs barrier via polynomial identity testing
• Mathematics, Computer Science
Electron. Colloquium Comput. Complex.
• 2017
It is observed that a certain kind of algebraic proof cannot be used to prove lower bounds against VP if and only if what the authors call succinct hitting sets exist for VP.

## References

SHOWING 1-10 OF 19 REFERENCES
Depth-3 arithmetic circuits over fields of characteristic zero
• Computer Science, Mathematics
computational complexity
• 2001
This paper proves quadratic lower bounds for depth-3 arithmetic circuits over fields of characteristic zero for the elementary symmetric functions, the (trace of) iterated matrix multiplication, and the determinant, and gives new shorter formulae of constant depth for the Elementary symmetrical functions.
Diagonal Circuit Identity Testing and Lower Bounds
In this paper we give the first deterministic polynomial time algorithm for testing whether a diagonaldepth-3 circuit C(x 1 ,...,x n ) (i.e. Cis a sum of powers of linear functions) is zero. We also
Lower Bounds and Separations for Constant Depth Multilinear Circuits
• Computer Science, Mathematics
Computational Complexity Conference
• 2008
An exponential lower bound for the size of constant depth multilinear arithmetic circuits computing either the determinant or the permanent is proved.
Exponential complexity lower bounds for depth 3 arithmetic circuits in algebras of functions over finite fields
• Computer Science, Mathematics
Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
• 1998
An exponential lower bound on the complexity of a depth 3 arithmetic circuit which computes the determinant or the permanent of a matrix considered as functions f:(F*)n/sup 2//spl rarr/F is proved.
Jacobian hits circuits: hitting-sets, lower bounds for depth-D occur-k formulas & depth-3 transcendence degree-k circuits
• Computer Science, Mathematics
STOC '12
• 2012
This work shows that polynomial time hitting-set generators for identity testing of the two seemingly different and well studied models - depth-3 circuits with bounded top fanin, and constant-depth constant-read multilinear formulas - can be constructed using one common algebraic-geometry theme: Jacobian captures algebraic independence.
Affine projections of symmetric polynomials
• Amir Shpilka
• Computer Science, Mathematics
Proceedings 16th Annual IEEE Conference on Computational Complexity
• 2001
It is shown that an answer of the following problem will imply lower bounds on symmetric circuits for many polynomials, and the main technical contribution relates the maximal dimension of linear subspaces on which S/sub m//sup d/ vanishes, and lower bounds to the symmetric model.
Arithmetic circuits: The chasm at depth four gets wider
• P. Koiran
• Mathematics, Computer Science
Theor. Comput. Sci.
• 2012
Lower bounds on arithmetic circuits via partial derivatives
• Mathematics, Computer Science
computational complexity
• 2005
A new technique is described for obtaining lower bounds on restricted classes of non-monotone arithmetic circuits, based on the linear span of their partial derivatives, for multivariate polynomials.
Arithmetic Circuits: A survey of recent results and open questions
• Computer Science, Mathematics
Found. Trends Theor. Comput. Sci.
• 2010
The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what it finds to be the most interesting and accessible research directions, with an emphasis on works from the last two decades.
Lower Bounds for Arithmetic Circuits via Partial Serivatives (Preliminary Version).
• Computer Science, Mathematics
FOCS 1995
• 1995
A new technique is described for obtaining lower bounds on restricted classes of non-monotone arithmetic circuits based on the linear span of their partial derivatives for multivariate polynomials and iterated matrix products.