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The problem of finding large cliques in random graphs and its " planted" variant, where one wants to recover a clique of size ω log (n) added to an Erd˝ os-Rényi graph G ∼ G(n, 1 2), have been intensely studied. Nevertheless, existing polynomial time algorithms can only recover planted cliques of size ω = Ω(√ n). By contrast, information theoretically, one… (More)

We study a statistical model for the tensor principal component analysis problem introduced by Montanari and Richard: Given a order-3 tensor T of the form T = τ·v ⊗3 0 +A, where τ 0 is a signal-to-noise ratio, v 0 is a unit vector, and A is a random noise tensor, the goal is to recover the planted vector v 0. For the case that A has iid standard Gaussian… (More)

We consider two problems that arise in machine learning applications: the problem of recovering a planted sparse vector in a random linear subspace and the problem of decomposing a random low-rank overcomplete 3-tensor. For both problems, the best known guarantees are based on the sum-of-squares method. We develop new algorithms inspired by analyses of the… (More)

We prove that with high probability over the choice of a random graph G from the Erdös-Rényi distribution G(n,1/2), the n<sup>O(d)</sup>-time degree d Sum-of-Squares semidefinite programming relaxation for the clique problem will give a value of at least n<sup>1/2-c(d/log n)1/2</sup> for some constant c > 0. This yields a nearly tight… (More)

We consider two problems that arise in machine learning applications: the problem of recovering a planted sparse vector in a random linear subspace and the problem of decomposing a random low-rank overcomplete 3-tensor. For both problems, the best known guarantees are based on the sum-of-squares method. We develop new algorithms inspired by analyses of the… (More)

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