Convergence bounds for empirical nonlinear least-squares

We consider best approximation problems in a nonlinear subset [[EQUATION]] of a Banach space of functions [[EQUATION]] . The norm is assumed to be a generalization of the [[EQUATION]] -norm for which only a weighted Monte Carlo estimate [[EQUATION]] can be computed. The objective is to obtain an approximation [[EQUATION]] of an unknown function [[EQUATION]] by minimizing the empirical norm [[EQUATION]] . We consider this problem for general nonlinear subsets and establish error bounds for the empirical best approximation error. Our results are based on a restricted isometry property (RIP) which holds in probability and is independent of the nonlinear least squares setting. Several model classes are examined where analytical statements can be made about the RIP and the results are compared to existing sample complexity bounds from the literature. We find that for well-studied model classes our general bound is weaker but exhibits many of the same properties as these specialized bounds. Notably, we demonstrate the advantage of an optimal sampling density (as known for linear spaces) for sets of functions with sparse representations.

Difference between two species of emu hides a test for lepton flavour violation

We argue that an LHC measurement of some simple quantities related to the ratio of rates of e + μ − to e − μ + events is surprisingly sensitive to as-yet unexcluded R-parity violating supersymmetric models with non-zero λ 231 ′ couplings. The search relies upon the approximate lepton universality in the Standard Model, the sign of the charge of the proton, and a collection of favourable detector biases. The proposed search is unusual because: it does not require any of the displaced vertices, hadronic neutralino decay products, or squark/gluino production relied upon by existing LHC RPV searches; it could work in cases in which the only light sparticles were smuons and neutralinos; and it could make a discovery (though not necessarily with optimal significance) without requiring the computation of a leading-order Monte Carlo estimate of any background rate. The LHC has shown no strong hints of post-Higgs physics and so precision Standard Model measurements are becoming ever more important. We argue that in this environment growing profits are to be made from searches that place detector biases and symmetries of the Standard Model at their core — searches based around ‘controls’ rather than around signals.

Bayesian optimal design of fixed knockout tournament brackets

We present a methodology for finding globally optimal knockout tournament designs when partial information is known about the strengths of the players. Our approach involves maximizing an expected utility through a Bayesian optimal design framework. Given the prohibitive computational barriers connected with direct computation, we compute a Monte Carlo estimate of the expected utility for a fixed tournament bracket, and optimize the expected utility through simulated annealing. We demonstrate our method by optimizing the probability that the best player wins the tournament. We compare our approach to other knockout tournament designs, including brackets following the standard seeding. We also demonstrate how our approach can be applied to a variety of other utility functions, including whether the best two players meet in the final, the consistency between the number of wins and the player strengths, and whether the players are matched up according to the standard seeding.

On the Behaviour of the Backward Interpretation of Feynman-Kac Formulae Under Verifiable Conditions

We consider the time behaviour associated to the sequential Monte Carlo estimate of the backward interpretation of Feynman-Kac formulae. This is particularly of interest in the context of performing smoothing for hidden Markov models. We prove a central limit theorem under weaker assumptions than adopted in the literature. We then show that the associated asymptotic variance expression for additive functionals grows at most linearly in time under hypotheses that are weaker than those currently existing in the literature. The assumptions are verified for some hidden Markov models.

On the Behaviour of the Backward Interpretation of Feynman-Kac Formulae Under Verifiable Conditions

We consider the time behaviour associated to the sequential Monte Carlo estimate of the backward interpretation of Feynman-Kac formulae. This is particularly of interest in the context of performing smoothing for hidden Markov models. We prove a central limit theorem under weaker assumptions than adopted in the literature. We then show that the associated asymptotic variance expression for additive functionals grows at most linearly in time under hypotheses that are weaker than those currently existing in the literature. The assumptions are verified for some hidden Markov models.