Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Near-Optimal Depth-Hierarchy Theorem for Small-Depth Multilinear
  Circuits   We study the size blow-up that is necessary to convert an algebraic circuit
of product-depth $\Delta+1$ to one of product-depth $\Delta$ in the multilinear
setting.
  We show that for every positive $\Delta = \Delta(n) = o(\log n/\log \log n),$
there is an explicit multilinear polynomial $P^{(\Delta)}$ on $n$ variables
that can be computed by a multilinear formula of product-depth $\Delta+1$ and
size $O(n)$, but not by any multilinear circuit of product-depth $\Delta$ and
size less than $\exp(n^{\Omega(1/\Delta)})$. This result is tight up to the
constant implicit in the double exponent for all $\Delta = o(\log n/\log \log
n).$
  This strengthens a result of Raz and Yehudayoff (Computational Complexity
2009) who prove a quasipolynomial separation for constant-depth multilinear
circuits, and a result of Kayal, Nair and Saha (STACS 2016) who give an
exponential separation in the case $\Delta = 1.$
  Our separating examples may be viewed as algebraic analogues of variants of
the Graph Reachability problem studied by Chen, Oliveira, Servedio and Tan
(STOC 2016), who used them to prove lower bounds for constant-depth Boolean
circuits.

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The categories of this paper are: comp-gas
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The categories of this paper are: hep-ex
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The categories of this paper are: astro-ph
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The categories of this paper are: patt-sol
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: q-bio
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The categories of this paper are: mtrl-th
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The categories of this paper are: atom-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-th
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The categories of this paper are: q-alg
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The categories of this paper are: math
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The categories of this paper are: alg-geom