## 3 Citations

On the Power of Homogeneous Depth 4 Arithmetic Circuits

- Computer Science, Mathematics2014 IEEE 55th Annual Symposium on Foundations of Computer Science
- 2014

It is shown that any homogeneous depth 4 circuit computing the (1, 1) entry in the product of n generic matrices of dimension nO(1) must have size nΩ(√n) and the results strengthen previous works in two significant ways.

An exponential lower bound for homogeneous depth-5 circuits over finite fields

- Mathematics, Computer ScienceElectron. Colloquium Comput. Complex.
- 2015

This paper shows exponential lower bounds for the class of homogeneous depth-$5 circuits over all small finite fields and builds over a tighter analysis of the lower bound of Kumar and Saraf [KS14], the first super-polynomial lower bound for this class for any field.

Lower bounds for bounded depth arithmetic circuits

- Computer Science, Mathematics
- 2017

A strong hierarchy theorem for bottom fan-in for homogeneous depth-4 circuits, a superpolynomial lower bound for homogeneity depth-5 circuits over finite fields, and some results indicating that the parameters for depth reduction to homogeneous Depth 4 circuits might be close to optimal in a fairly strong sense.

## References

SHOWING 1-10 OF 26 REFERENCES

Improved bounds for reduction to depth 4 and depth 3

- Computer Science, MathematicsInf. Comput.
- 2015

On the Power of Homogeneous Depth 4 Arithmetic Circuits

- Computer Science, Mathematics2014 IEEE 55th Annual Symposium on Foundations of Computer Science
- 2014

It is shown that any homogeneous depth 4 circuit computing the (1, 1) entry in the product of n generic matrices of dimension nO(1) must have size nΩ(√n) and the results strengthen previous works in two significant ways.

Lower bounds for depth 4 formulas computing iterated matrix multiplication

- Mathematics, Computer ScienceSTOC
- 2014

It is shown that any multilinear homogeneous depth 4 arithmetic formula computing the product of d generic matrices of size n × n, IMMn,d, has size nΩ(√d) as long as d = nO(1), improving the result of Nisan and Wigderson (Computational Complexity, 1997).

Arithmetic Circuit Lower Bounds via MaxRank

- Computer Science, MathematicsICALP
- 2013

The polynomial coefficient matrix is introduced and maximum rank of this matrix under variable substitution is identified as a complexity measure for multivariate polynomials and super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits are proved.

An Exponential Lower Bound for Homogeneous Depth Four Arithmetic Formulas

- Computer Science2014 IEEE 55th Annual Symposium on Foundations of Computer Science
- 2014

This work gives an explicit family of polynomials of degree d on N variables with 0, 1-coefficients such that for any representation of a polynomial f in this family of the form f = Σi,j, it must hold that 2Ω(√d·log N) ≥ 2.

Arithmetic circuits: The chasm at depth four gets wider

- Mathematics, Computer ScienceTheor. Comput. Sci.
- 2012

A super-polynomial lower bound for regular arithmetic formulas

- Mathematics, Computer ScienceSTOC
- 2013

It is shown that there is an n2-variate polynomial of degree n in VNP such that any regular formula computing it must be of size at least nΩ(log n) and any further asymptotic improvement in the lower bound for such formulas will imply that VP is different from VNP.

Depth-4 Lower Bounds, Determinantal Complexity: A Unified Approach

- Computer Science, Mathematicscomputational complexity
- 2019

A simple combinatorial property is identified such that any polynomial f that satisfies this property would achieve a similar depth-4 circuit-size lower bound and it does not matter whether f is in VNP or in $$\mathsf {VNP}$$VNP, which gets a simple unified lower-bound analysis for the above-mentioned polynomials.

Diagonal Circuit Identity Testing and Lower Bounds

- Computer Science, MathematicsICALP
- 2007

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…

Depth-3 arithmetic circuits over fields of characteristic zero

- Computer Science, Mathematicscomputational 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.