Denoising, detrending, deconvolution: usual restoration tasks, traditionally decoupled. Coupled formulations entail complex ill-posed inverse problems. We propose PENDANTSS for joint trend removal and blind deconvolution of sparse peak-like signals. It blends a parsimonious prior with the hypothesis that smooth trend and noise can somewhat be separated by low-pass filtering. We combine the generalized quasi-norm ratio SOOT/SPOQ sparse penalties $\ell_p/\ell_q$ with the BEADS ternary assisted source separation algorithm. This results in a both convergent and efficient tool, with a novel Trust-Region block alternating variable metric forward-backward approach. It outperforms comparable methods, when applied to typically peaked analytical chemistry signals. Reproducible code is provided.

Relating a set of variables X to a response y is crucial in chemometrics. A quantitative prediction objective can be enriched by qualitative data interpretation, for instance by locating the most influential features. When high-dimensional problems arise, dimension reduction techniques can be used. Most notable are projections (e.g. Partial Least Squares or PLS ) or variable selections (e.g. lasso). Sparse partial least squares combine both strategies, by blending variable selection into PLS. The variant presented in this paper, Dual-sPLS, generalizes the classical PLS1 algorithm. It provides balance between accurate prediction and efficient interpretation. It is based on penalizations inspired by classical regression methods (lasso, group lasso, least squares, ridge) and uses the dual norm notion. The resulting sparsity is enforced by an intuitive shrinking ratio parameter. Dual-sPLS favorably compares to similar regression methods, on simulated and real chemical data. Code is provided as an open-source package in R: \url{https://CRAN.R-project.org/package=dual.spls}.

We propose a method to reconstruct sparse signals degraded by a nonlinear distortion and acquired at a limited sampling rate. Our method formulates the reconstruction problem as a nonconvex minimization of the sum of a data fitting term and a penalization term. In contrast with most previous works which settle for approximated local solutions, we seek for a global solution to the obtained challenging nonconvex problem. Our global approach relies on the so-called Lasserre relaxation of polynomial optimization. We here specifically include in our approach the case of piecewise rational functions, which makes it possible to address a wide class of nonconvex exact and continuous relaxations of the $\ell_0$ penalization function. Additionally, we study the complexity of the optimization problem. It is shown how to use the structure of the problem to lighten the computational burden efficiently. Finally, numerical simulations illustrate the benefits of our method in terms of both global optimality and signal reconstruction.