# Improved bounds for reduction to depth 4 and depth 3

@article{Tavenas2013ImprovedBF,
title={Improved bounds for reduction to depth 4 and depth 3},
author={S{\'e}bastien Tavenas},
journal={Inf. Comput.},
year={2013},
volume={240},
pages={2-11}
}
107 Citations
Finer separations between shallow arithmetic circuits
• Mathematics, Computer Science
Electron. Colloquium Comput. Complex.
• 2016
The proofs are much shorter and simpler than the two known proofs of n^{Omega(sqrt(d))} lower bound for homogeneous depth-4 circuits, albeit the proofs only work when d = O(log^2(n), which shows that the parameters of depth reductions are optimal for algebraic branching programs.
On the Power of Border of Depth-3 Arithmetic Circuits
• Mrinal Kumar
• Mathematics, Computer Science
ACM Trans. Comput. Theory
• 2020
If a degree d homogeneous polynomial P can be computed by an arithmetic circuit of size s ≥ d, then for every t ≤ d, P is in the border of a depth-3 circuit of top fan-in sO (t) and formal degree sO(d/t).
Almost Cubic Bound for Depth Three Circuits in VP
• Morris Yau
• Computer Science, Mathematics
Electron. Colloquium Comput. Complex.
• 2016
It is shown that for every N and D satisfying poly(N) > D > log2 N, there exist polynomials PN,D on N variable of degree D in VP that can not be computed by circuits of size Ω̃(N2D).
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.
Depth-4 Lower Bounds, Determinantal Complexity: A Unified Approach
• Computer Science, Mathematics
computational 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.
Towards Optimal Depth Reductions for Syntactically Multilinear Circuits
• Computer Science
Electron. Colloquium Comput. Complex.
• 2019
It follows from the lower bounds of Raz and Yehudayoff (CC 2009) that in general, for constant $\Delta$, the exponent in this upper bound is tight and cannot be improved to $o\left(n/\log n)^{1/\Delta}\cdot \log n\right)$.
A super-quadratic lower bound for depth four arithmetic circuits
• Computer Science, Mathematics
Electron. Colloquium Comput. Complex.
• 2020
An Ω(n2.5) lower bound is shown for general depth four arithmetic circuits computing an explicit n-variate degree-Θ(n) multilinear polynomial over any field of characteristic zero, inspired by a well-known greedy approximation algorithm for the weighted set-cover problem.
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.
Arithmetic Circuits with Locally Low Algebraic Rank
• Mathematics, Computer Science
Theory Comput.
• 2017
A key technical ingredient of the proofs, which may be of independent interest, is a result which states that over any field of characteristic zero, up to a translation, every polynomial in a set of polynomials can be written as a function of the polynOMials in a transcendence basis of the set.

## References

SHOWING 1-10 OF 13 REFERENCES
Arithmetic Circuits: A Chasm at Depth Three
• Computer Science, Mathematics
2013 IEEE 54th Annual Symposium on Foundations of Computer Science
• 2013
We show that, over Q, if an n-variate polynomial of degree d = n<sup>O(1)</sup> is computable by an arithmetic circuit of size s (respectively by an arithmetic branching program of size s) then it
Lower bounds for depth 4 formulas computing iterated matrix multiplication
• Mathematics, Computer Science
STOC
• 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 circuits: The chasm at depth four gets wider
• P. Koiran
• Mathematics, Computer Science
Theor. Comput. Sci.
• 2012
Non-Commutative Arithmetic Circuits: Depth Reduction and Size Lower Bounds
• Mathematics, Computer Science
Theor. Comput. Sci.
• 1998
Approaching the Chasm at Depth Four
• Computer Science
2013 IEEE Conference on Computational Complexity
• 2013
Any homogeneous depth four arithmetic circuit with bottom fanin bounded by √n computing the permanent (or the determinant) must be of size exp(Ω(√n)).
Characterizing Valiant's algebraic complexity classes
• Mathematics, Computer Science
J. Complex.
• 2008
Partial Derivatives in Arithmetic Complexity and Beyond (Foundations and Trends(R) in Theoretical Computer Science)
• Mathematics, Computer Science
• 2011
Partial Derivatives in Arithmetic Complexity and Beyond goes on to look at applications which go beyond computational complexity, where partial derivatives provide a wealth of structural information about polynomials, and help us solve various number theoretic, geometric, and combinatorial problems.
Fast Parallel Computation of Polynomials Using Few Processors
• Computer Science
SIAM J. Comput.
• 1983
It is shown that any multivariate polynomial of degree d that can be compute sequentially in C steps can be computed in parallel in O(1) using only \$(Cd)^{O(1)} processors.
Partial Derivatives in Arithmetic Complexity and Beyond
• Mathematics, Computer Science
Found. Trends Theor. Comput. Sci.
• 2011
The bulk of the survey shows that partial derivatives provide essential ingredients in proving both upper and lower bounds for computing polynomials by a variety of natural arithmetic models.
Completeness and Reduction in Algebraic Complexity Theory
The structure of Valiant's Algebraic Model of NP-Completeness is illustrated with some Complete Families of Polynomials and the Complexity of Immanants.