arXiv subjects

Samuel B. Hopkins

Publications and source records attributed to Samuel B. Hopkins.

39 records · Page 3Linked to original sources

arXiv2016

Fast spectral algorithms from sum-of-squares proofs: tensor decomposition and planted sparse vectors

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 sum-of-squares method. Our algorithms achieve the same or similar guarantees as sum-of-squares for these problems but the running time is significantly faster. For the planted sparse vector problem, we give an algorithm with running time nearly linear in the input size that approximately recovers a planted sparse vector with up to constant relative sparsity in a random subspace of $\mathbb R^n$ of dimension up to $\tilde Ω(\sqrt n)$. These recovery guarantees match the best known ones of Barak, Kelner, and Steurer (STOC 2014) up to logarithmic factors. For tensor decomposition, we give an algorithm with running time close to linear in the input size (with exponent $\approx 1.086$) that approximately recovers a component of a random 3-tensor over $\mathbb R^n$ of rank up to $\tilde Ω(n^{4/3})$. The best previous algorithm for this problem due to Ge and Ma (RANDOM 2015) works up to rank $\tilde Ω(n^{3/2})$ but requires quasipolynomial time.

cs.DS↗

arXiv2015

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 $ω\gg \log{(n)}$ 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 $ω= Ω(\sqrt{n})$. By contrast, information theoretically, one can recover planted cliques so long as $ω\gg \log{(n)}$. 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 $ω\gg \sqrt n poly \log n$, improving upon the bound $ω\gg n^{1/3}$ 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 $ω\gg n^{1/(d + 1)} /(2^d poly \log n)$, 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.

cs.CC↗

arXiv2015

Tensor principal component analysis via sum-of-squares proofs

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 $T = τ\cdot v_0^{\otimes 3} + A$, where $τ\geq 0$ 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 entries, we give an efficient algorithm to recover $v_0$ whenever $τ\geq ω(n^{3/4} \log(n)^{1/4})$, and certify that the recovered vector is close to a maximum likelihood estimator, all with high probability over the random choice of $A$. The previous best algorithms with provable guarantees required $τ\geq Ω(n)$. In the regime $τ\leq o(n)$, natural tensor-unfolding-based spectral relaxations for the underlying optimization problem break down (in the sense that their integrality gap is large). To go beyond this barrier, we use convex relaxations based on the sum-of-squares method. Our recovery algorithm proceeds by rounding a degree-$4$ sum-of-squares relaxations of the maximum-likelihood-estimation problem for the statistical model. To complement our algorithmic results, we show that degree-$4$ sum-of-squares relaxations break down for $τ\leq O(n^{3/4}/\log(n)^{1/4})$, which demonstrates that improving our current guarantees (by more than logarithmic factors) would require new techniques or might even be intractable. Finally, we show how to exploit additional problem structure in order to solve our sum-of-squares relaxations, up to some approximation, very efficiently. Our fastest algorithm runs in nearly-linear time using shifted (matrix) power iteration and has similar guarantees as above. The analysis of this algorithm also confirms a variant of a conjecture of Montanari and Richard about singular vectors of tensor unfoldings.

cs.LG↗