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

Parliamentary functions portrayed on European parliaments' websites

This article uses data from a pilot study on the websites of 15 parliaments in Europe to identify which parliamentary functions are portrayed on these websites. The pilot study undertook a contents analysis focused on four parliamentary functions: legislation, legitimation, representation and scrutiny, to ascertain the extent to which each of these functions is present on the websites analysed. As parliaments face difficulties of public perception, their websites become an important tool for dissemination of parliamentary work. And yet we know little about what these websites portray, when it comes to parliamentary activity. Are these websites a reflection of parliamentary work? To what extent do these websites express political debate? To what extent are these websites comprehensive to all of the roles performed by parliaments? This article shows that legislation is the main focus of parliamentary websites in Europe and representation the one that has less devoted space. This may go some way to explain why some parliamentary websites are considered as too dense and not user friendly. We also show that parliamentary websites tend to focus on parliamentary outputs to the detriment of parliamentary actors. Because of their institutional collective representation, parliaments are cautious in focusing on politicians or parties; instead they focus on what parliaments produce (laws, questions, debates, etc.). Again, this goes some way to explain why parliaments' websites may not be as engaging as one may expect, simply because they are not meant to reflect political debate, but simply to facilitate it

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 $X$ is the $n\times n$ symmetric matrix with $X_{ij}\sim N(0,b_{ij}^2)$, we show that $$\mathbf{E}\Vert X\Vert \lesssim\max_i\sqrt{\sum_jb_{ij}^2}+\max _{ij}\vert b_{ij}\vert \sqrt{\log n}.$$ 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

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