Samuel B. Hopkins

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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, v0 is a unit vector, and A is a random noise tensor, the goal is to recover the planted vector v0. For the case that A has iid standard Gaussian(More)
We prove that with high probability over the choice of a random graph G from the Erdo&#x0308;s-Re&#x0301;nyi 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 &gt; 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)
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 Erdős-Rényi graph G ∼ G(n, 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 can(More)
We consider two problems that arise in machine learning applications: the problem of recovering aplanted sparse vector in a random linear subspace and theproblemofdecomposing 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 propose an efficient meta-algorithm for Bayesian estimation problems that is based on low-degree polynomials, semidefinite programming, and tensor decomposition. The algorithm is inspired by recent lower bound constructions for sum-of-squares and related to the method of moments. Our focus is on sample complexity bounds that are as tight as possible (up(More)
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