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تسجيل مستخدم جديدImproved Sum-of-Squares Lower Bounds for Hidden Clique and Hidden Submatrix Problems
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Given a large data matrix $Ainmathbb{R}^{ntimes n}$, we consider the problem of determining whether its entries are i.i.d. with some known marginal distribution $A_{ij}sim P_0$, or instead $A$ contains a principal submatrix $A_{{sf Q},{sf Q}}$ whose entries have marginal distribution $A_{ij}sim P_1 eq P_0$. As a special case, the hidden (or planted) clique problem requires to find a planted clique in an otherwise uniformly random graph. Assuming unbounded computational resources, this hypothesis testing problem is statistically solvable provided $|{sf Q}|ge C log n$ for a suitable constant $C$. However, despite substantial effort, no polynomial time algorithm is known that succeeds with high probability when $|{sf Q}| = o(sqrt{n})$. Recently Meka and Wigderson cite{meka2013association}, proposed a method to establish lower bounds within the Sum of Squares (SOS) semidefinite hierarchy. Here we consider the degree-$4$ SOS relaxation, and study the construction of cite{meka2013association} to prove that SOS fails unless $kge C, n^{1/3}/log n$. An argument presented by Barak implies that this lower bound cannot be substantially improved unless the witness construction is changed in the proof. Our proof uses the moments method to bound the spectrum of a certain random association scheme, i.e. a symmetric random matrix whose rows and columns are indexed by the edges of an Erdos-Renyi random graph.
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Finding cliques in random graphs and the closely related planted clique variant, where a clique of size t is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best known polynom
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We prove that for every $epsilon>0$ and predicate $P:{0,1}^krightarrow {0,1}$ that supports a pairwise independent distribution, there exists an instance $mathcal{I}$ of the $mathsf{Max}P$ constraint satisfaction problem on $n$ variables such that no
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We prove that with high probability over the choice of a random graph $G$ from the ErdH{o}s-Renyi distribution $G(n,1/2)$, the $n^{O(d)}$-time degree $d$ Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of a
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Several works have shown unconditional hardness (via integrality gaps) of computing equilibria using strong hierarchies of convex relaxations. Such results however only apply to the problem of computing equilibria that optimize a certain objective fu
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