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}
}
On the limits of depth reduction at depth 3 over small finite fields
Finer separations between shallow arithmetic circuits
TLDR
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
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    ACM Trans. Comput. Theory
  • 2020
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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
TLDR
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.
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