Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: SoS and Planted Clique: Tight Analysis of MPW Moments at all Degrees and
  an Optimal Lower Bound at Degree Four   The problem of finding large cliques in random graphs and its "planted"
variant, where one wants to recover a clique of size $\omega \gg \log{(n)}$
added to an \Erdos-\Renyi graph $G \sim G(n,\frac{1}{2})$, have been intensely
studied. Nevertheless, existing polynomial time algorithms can only recover
planted cliques of size $\omega = \Omega(\sqrt{n})$. By contrast, information
theoretically, one can recover planted cliques so long as $\omega \gg
\log{(n)}$. In this work, we continue the investigation of algorithms from the
sum of squares hierarchy for solving the planted clique problem begun by Meka,
Potechin, and Wigderson (MPW, 2015) and Deshpande and Montanari (DM,2015). Our
main results improve upon both these previous works by showing:
  1. Degree four SoS does not recover the planted clique unless $\omega \gg
\sqrt n poly \log n$, improving upon the bound $\omega \gg n^{1/3}$ due to DM.
A similar result was obtained independently by Raghavendra and Schramm (2015).
  2. For $2 < d = o(\sqrt{\log{(n)}})$, degree $2d$ SoS does not recover the
planted clique unless $\omega \gg n^{1/(d + 1)} /(2^d poly \log n)$, improving
upon the bound due to MPW.
  Our proof for the second result is based on a fine spectral analysis of the
certificate used in the prior works MPW,DM and Feige and Krauthgamer (2003) by
decomposing it along an appropriately chosen basis. Along the way, we develop
combinatorial tools to analyze the spectrum of random matrices with dependent
entries and to understand the symmetries in the eigenspaces of the set
symmetric matrices inspired by work of Grigoriev (2001).
  An argument of Kelner shows that the first result cannot be proved using the
same certificate. Rather, our proof involves constructing and analyzing a new
certificate that yields the nearly tight lower bound by "correcting" the
certificate of previous works.

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The categories of this paper are: solv-int
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The categories of this paper are: chao-dyn
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The categories of this paper are: alg-geom
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The categories of this paper are: eess
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The categories of this paper are: q-alg
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The categories of this paper are: atom-ph
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The categories of this paper are: q-fin
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The categories of this paper are: hep-lat
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The categories of this paper are: hep-th
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The categories of this paper are: ao-sci
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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: astro-ph
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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