In Nuclear Magnetic Resonance (NMR) spectroscopy, an efficient analysis and a relevant extraction of different molecule properties from a given chemical mixture are important tasks, especially when processing bidimensional NMR data. To that end, using a blind source separation approach based on a variational formulation seems to be a good strategy. However, the poor resolution of NMR spectra and their large dimension require a new and modern blind source separation method. In this work, we propose a new variational formulation for blind source separation (BSS) based on a β-divergence data fidelity term combined with sparsity promoting regularization functions. An application to 2D HSQC NMR experiments illustrates the interest and the effectiveness of the proposed method whether in simulated or real cases.

We study the properties of the Frank-Wolfe algorithm to solve the m-EXACT-SPARSE reconstruction problem, where a signal y must be expressed as a sparse linear combination of a predefined set of atoms, called dictionary. We prove that when the signal is sparse enough with respect to the coherence of the dictionary, then the iterative process implemented by the Frank-Wolfe algorithm only recruits atoms from the support of the signal, is the smallest set of atoms from the dictionary that allows for a perfect reconstruction of y. We also prove that under this same condition, there exists an iteration beyond which the algorithm converges exponentially.

We present a generalization of Subspace Pursuit, which seeks the fc-sparse vector that minimizes a generic cost function. We introduce the Restricted Diagonal Property, which much like RIP in the classical setting, enables to control the convergence of Generalized Subspace Pursuit (GSP). To tackle the problem of Poisson denoising, we propose to use GSP together with the Moreau-Yosida approximation of the Poisson likelihood. Experiments were conducted on synthetic, exact sparse and natural images corrupted by Poisson noise. We study the influence of the different parameters and show that our approach performs better than Subspace Pursuit or ℓ1-relaxed methods and compares favorably to state-of-art methods.

Cone Beam Computerized Tomography (CBCT) and Positron Emission Tomography (PET) Scans are medical imaging devices that require solving ill-posed inverse problems. The models considered come directly from the physics of the acquisition devices, and take into account the specificity of the (Poisson) noise. We propose various fast numerical schemes to compute the solution. In particular, we show that a new algorithm recently introduced by A. Chambolle and T. Pock is well suited in the PET case when considering non differentiable regularizations such as total variation or wavelet ℓ1-regularization. Numerical experiments indicate that the proposed algorithms compare favorably with respect to well-established methods in tomography.

We present original empirical Bernstein inequalities for U-statistics with bounded symmetric kernels q. They are expressed with respect to empirical estimates of either the variance of q or the conditional variance that appears in the Bernstein-type inequality for U-statistics derived by Arcones [2]. Our result subsumes other existing empirical Bernstein inequalities, as it reduces to them when U-statistics of order 1 are considered. In addition, it is based on a rather direct argument using two applications of the same (non-empirical) Bernstein inequality for U-statistics. We discuss potential applications of our new inequalities, especially in the realm of learning ranking/scoring functions. In the process, we exhibit an efficient procedure to compute the variance estimates for the special case of bipartite ranking that rests on a sorting argument. We also argue that our results may provide test set bounds and particularly interesting empirical racing algorithms for the problem of online learning of scoring functions.