2,016 research outputs found
A note on Probably Certifiably Correct algorithms
Many optimization problems of interest are known to be intractable, and while
there are often heuristics that are known to work on typical instances, it is
usually not easy to determine a posteriori whether the optimal solution was
found. In this short note, we discuss algorithms that not only solve the
problem on typical instances, but also provide a posteriori certificates of
optimality, probably certifiably correct (PCC) algorithms. As an illustrative
example, we present a fast PCC algorithm for minimum bisection under the
stochastic block model and briefly discuss other examples
Sharp nonasymptotic bounds on the norm of random matrices with independent entries
We obtain nonasymptotic bounds on the spectral norm of random matrices with
independent entries that improve significantly on earlier results. If is
the symmetric matrix with , we show that
This bound is optimal in the sense that a matching
lower bound holds under mild assumptions, and the constants are sufficiently
sharp that we can often capture the precise edge of the spectrum. Analogous
results are obtained for rectangular matrices and for more general sub-Gaussian
or heavy-tailed distributions of the entries, and we derive tail bounds in
addition to bounds on the expected norm. The proofs are based on a combination
of the moment method and geometric functional analysis techniques. As an
application, we show that our bounds immediately yield the correct phase
transition behavior of the spectral edge of random band matrices and of sparse
Wigner matrices. We also recover a result of Seginer on the norm of Rademacher
matrices.Comment: Published at http://dx.doi.org/10.1214/15-AOP1025 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Computational Hardness of Certifying Bounds on Constrained PCA Problems
Given a random n×n symmetric matrix W drawn from the Gaussian orthogonal ensemble (GOE), we consider the problem of certifying an upper bound on the maximum value of the quadratic form x⊤Wx over all vectors x in a constraint set S⊂Rn. For a certain class of normalized constraint sets S we show that, conditional on certain complexity-theoretic assumptions, there is no polynomial-time algorithm certifying a better upper bound than the largest eigenvalue of W. A notable special case included in our results is the hypercube S={±1/n−−√}n, which corresponds to the problem of certifying bounds on the Hamiltonian of the Sherrington-Kirkpatrick spin glass model from statistical physics.
Our proof proceeds in two steps. First, we give a reduction from the detection problem in the negatively-spiked Wishart model to the above certification problem. We then give evidence that this Wishart detection problem is computationally hard below the classical spectral threshold, by showing that no low-degree polynomial can (in expectation) distinguish the spiked and unspiked models. This method for identifying computational thresholds was proposed in a sequence of recent works on the sum-of-squares hierarchy, and is believed to be correct for a large class of problems. Our proof can be seen as constructing a distribution over symmetric matrices that appears computationally indistinguishable from the GOE, yet is supported on matrices whose maximum quadratic form over x∈S is much larger than that of a GOE matrix.ISSN:1868-896
Open problem: Tightness of maximum likelihood semidefinite relaxations
We have observed an interesting, yet unexplained, phenomenon: Semidefinite
programming (SDP) based relaxations of maximum likelihood estimators (MLE) tend
to be tight in recovery problems with noisy data, even when MLE cannot exactly
recover the ground truth. Several results establish tightness of SDP based
relaxations in the regime where exact recovery from MLE is possible. However,
to the best of our knowledge, their tightness is not understood beyond this
regime. As an illustrative example, we focus on the generalized Procrustes
problem
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