Michele Piana

Are loss functions all the same

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.

Projected Landweber method and preconditioning

The projected Landweber method is an iterative method for solving constrained least-squares problems when the constraints are expressed in terms of a convex and closed set . The convergence properties of the method have been recently investigated. Moreover, it has important applications to many problems of signal processing and image restoration. The practical difficulty is that the convergence is too slow. In this paper we apply to this method the so-called preconditioning which is frequently used for increasing the efficiency of the conjugate gradient method. We discuss the significance of preconditioning in this case and we show that it implies a modification of the original constrained least-squares problem. However, when the original problem is ill-posed, the approximate solutions provided by the preconditioned method are similar to those provided by the standard method if the preconditioning is suitably chosen. Moreover, the number of iterations can be reduced by a factor of 10 and even more. A few applications to problems of image restoration are also discussed.

Deducing Electron Properties from Hard X-Ray Observations

X-radiation from energetic electrons is the prime diagnostic of flare-accelerated electrons. The observed X-ray flux (and polarization state) is fundamentally a convolution of the cross-section for the hard X-ray emission process(es) in question with the electron distribution function, which is in turn a function of energy, direction, spatial location and time. To address the problems of particle propagation and acceleration one needs to infer as much information as possible on this electron distribution function, through a deconvolution of this fundamental relationship. This review presents recent progress toward this goal using spectroscopic, imaging and polarization measurements, primarily from the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI). Previous conclusions regarding the energy, angular (pitch angle) and spatial distributions of energetic electrons in solar flares are critically reviewed. We discuss the role and the observational evidence of several radiation processes: free-free electron-ion, free-free electron-electron, free-bound electron-ion, photoelectric absorption and Compton backscatter (albedo), using both spectroscopic and imaging techniques. This unprecedented quality of data allows for the first time inference of the angular distributions of the X-ray-emitting electrons and improved model-independent inference of electron energy spectra and emission measures of thermal plasma. Moreover, imaging spectroscopy has revealed hitherto unknown details of solar flare morphology and detailed spectroscopy of coronal, footpoint and extended sources in flaring regions. Additional attempts to measure hard X-ray polarization were not sufficient to put constraints on the degree of anisotropy of electrons, but point to the importance of obtaining good quality polarization data in the future.

Regularized electron flux spectra in the 2002 July 23 solar flare

By inverting the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI) hard X-ray photon spectrum with the Tikhonov regularization algorithm, we infer the effective mean electron source spectrum for a time interval near the peak of the 2002 July 23 event. This inverse approach yields the smoothest electron flux spectrum consistent with the data while retaining real features, such as local minima, that cannot be found with forward model-fitting methods that involve only a few parameters. A significant dip in the recovered mean source electron spectrum near E = 55 keV is noted, and its significance briefly discussed.

Numerical validation of the linear sampling method

The problem of visualizing scattering objects from far-field data can be addressed by a simple method, named linear sampling method (LSM), which requires the solution of ill-conditioned linear systems. In the present paper we perform a computational and experimental validation of the method, which is implemented by means of four different regularization algorithms. The effectiveness of the LSM when coupled with these algorithms is tested in the case of both simulated and real data. Furthermore a criterion for the choice of a level curve optimally approximating the profile of the scatterers is provided.

The linear sampling method without sampling

We present a new implementation of the linear sampling method in which the set of discretized far-field equations for all sampling points is replaced by a single-functional equation formulated in a Hilbert space defined as a direct sum of L2 spaces. The squared norm of the regularized solution of such an equation is used as an indicator function and is analytically determined together with its Fourier transform. This provides some theoretical hints about the spatial resolution achievable by the method.

GENERALIZED REGULARIZATION TECHNIQUES WITH CONSTRAINTS FOR THE ANALYSIS OF SOLAR BREMSSTRAHLUNG X-RAY SPECTRA

Hard X-ray spectra in solar flares provide knowledge of the electron spectrum that results from acceleration and propagation in the solar atmosphere. However, the inference of the electron spectra from solar X-ray spectra is an ill-posed inverse problem. Here, we develop and apply an enhanced regularization algorithm for this process making use of physical constraints on the form of the electron spectrum. The algorithm incorporates various features not heretofore employed in the solar flare context: Generalized Singular Value Decomposition (GSVD) to deal with different orders of constraints; rectangular form of the cross-section matrix to extend the solution energy range; regularization with various forms of the smoothing operator; and “preconditioning” of the problem. We show by simulations that this technique yields electron spectra with considerably more information and higher quality than previous algorithms.

Dynamical MEG Source Modeling with Multi-Target Bayesian Filtering

We present a Bayesian filtering approach for automatic estimation of dynamical source models from magnetoencephalographic data. We apply multi‐target Bayesian filtering and the theory of Random Finite Sets in an algorithm that recovers the life times, locations and strengths of a set of dipolar sources. The reconstructed dipoles are clustered in time and space to associate them with sources. We applied this new method to synthetic data sets and show here that it is able to automatically estimate the source structure in most cases more accurately than either traditional multi‐dipole modeling or minimum current estimation performed by uninformed human operators. We also show that from real somatosensory evoked fields the method reconstructs a source constellation comparable to that obtained by multi‐dipole modeling. Hum Brain Mapp, 2009.

Determination of electron flux spectra in a solar flare with an augmented regularization method: application to RHESSI data

Kontar et al. (2004) have shown how to recover mean source electron spectra