In this letter, we investigate the impact of choosing different loss functions from the viewpoint of statistical learning theory. We introduce a convexity assumption, which is met by all loss functions commonly used in the literature, and study how the bound on the estimation error changes with the loss. We also derive a general result on the minimizer of the expected risk for a convex loss function in the case of classification. The main outcome of our analysis is that for classification, the hinge loss appears to be the loss of choice. Other things being equal, the hinge loss leads to a convergence rate practically indistinguishable from the logistic loss rate and much better than the square loss rate. Furthermore, if the hypothesis space is sufficiently rich, the bounds obtained for the hinge loss are not loosened by the thresholding stage.

We survey the linear sampling method for solving the inverse scattering problem for time-harmonic electromagnetic waves at fixed frequency. We consider scattering by an obstacle as well as scattering by an inhomogeneous medium both in R2 and R3 .I ncl uded in our discussion is the use of regularization methods for ill-posed problems and numerical examples in both two and three dimensions. (Some figures in this article are in colour only in the electronic version) Electromagnetic scattering is a physical phenomenon whereby, in the presence of an inhomogeneity, an electromagnetic incident wave is scattered and the total field at any point in space can be written as the sum of the original incident field and the scattered field. The electromagnetic direct scattering problem is the problem of determining the scattered field when th eg eometrical and physical properties of the scatterer are known. The inverse scattering problem is the problem of inferring information on the inhomogeneity from knowledge of the far-field pattern, i.e. the scattered wave at large distances from the scatterer. Classical electromagnetic inverse scattering theory provides a rich supply of challenging mathematical problems. First of all, this is due to the fact that the unknown physical conditions where the scattering takes place create difficulties in trying to construct the solution procedure. Second, all the inverse scattering problems significant in applications belong to the class of so-called ill-posed problems in the sense of Hadamard (25) and therefore any reliable approach to their solution must face, at some stage, questions of uniqueness and numerical stability. In particular, it is well established that any numerical implementation of a method for the solution of an inverse scattering problem must, at some step, incorporate a regularization procedure in order to eliminate the artificial oscillations due to the ill-posedness of the problem.

The European Union funded the FLARECAST project, that ran from January 2015 until February 2018. FLARECAST had a research-to-operations (R2O) focus, and accordingly introduced several innovations into the discipline of solar flare forecasting. FLARECAST innovations were: first, the treatment of hundreds of physical properties viewed as promising flare predictors on equal footing, extending multiple previous works; second, the use of fourteen (14) different machine learning techniques, also on equal footing, to optimize the immense Big Data parameter space created by these many predictors; third, the establishment of a robust, three-pronged communication effort oriented toward policy makers, space-weather stakeholders and the wider public. FLARECAST pledged to make all its data, codes and infrastructure openly available worldwide. The combined use of 170+ properties (a total of 209 predictors are now available) in multiple machine-learning algorithms, some of which were designed exclusively for the project, gave rise to changing sets of best-performing predictors for the forecasting of different flaring levels, at least for major flares. At the same time, FLARECAST reaffirmed the importance of rigorous training and testing practices to avoid overly optimistic pre-operational prediction performance. In addition, the project has (a) tested new and revisited physically intuitive flare predictors and (b) provided meaningful clues toward the transition from flares to eruptive flares, namely, events associated with coronal mass ejections (CMEs). These leads, along with the FLARECAST data, algorithms and infrastructure, could help facilitate integrated space-weather forecasting efforts that take steps to avoid effort duplication. In spite of being one of the most intensive and systematic flare forecasting efforts to-date, FLARECAST has not managed to convincingly lift the barrier of stochasticity in solar flare occurrence and forecasting: solar flare prediction thus remains inherently probabilistic.