Inspired by several real-life applications in audio processing and medical image analysis, where the quantity of interest is generated by several sources to be accurately modeled and separated, as well as by recent advances in  regularization theory and optimization, we study the conditions on optimal support recovery in inverse problems of unmixing type by means of multi-penalty regularization.  

We consider and analyze a regularization functional composed of a data-fidelity term, where signal and noise are additively mixed, a non-smooth, convex, sparsity promoting term, and a quadratic penalty term to model the noise. We prove not only that  the well-established theory for sparse recovery in the single parameter case can be translated to the multi-penalty settings, but  we also demonstrate the enhanced properties of multi-penalty regularization in terms of support identification compared to sole $\ell^1$-minimization.  We additionally confirm and support the theoretical results by extensive numerical simulations, which give a statistics of robustness of the multi-penalty regularization scheme with respect to the single-parameter counterpart. 
Eventually, we confirm a significant improvement in performance compared to standard $\ell^1$-regularization for compressive sensing problems considered in our experiments. 

In real-world image processing applications, the data is high dimensional but the amount of high-quality data needed to train the model is very limited. In this paper, we extend a recently presented method for dictionary learning from incomplete data, the so-called Iterative Thresholding and $K$ residual Means for Masked data, to deal with high-dimensional data in an efficient way. In particular, the proposed algorithm incorporates a corruption model directly at the dictionary learning stage, also enabling reconstruction of the low-rank component again from corrupted signals. These modifications circumvent some difficulties associated with the efficient dictionary learning procedure in the presence of limited or incomplete data.

We choose an image inpainting problem as a guiding example, and further propose a procedure for automatic detection and reconstruction of the low-rank component from incomplete data and adaptive parameter selection for the sparse image reconstruction. We benchmark the efficacy and efficiency of our algorithm in terms of computing time and accuracy on colour and 3D medical images by comparing it to its dictionary learning counterparts.

In this paper we propose a new method for detecting relevant variables
from a priori given high-dimensional data under the assumption that input-
output dependence is described by a nonlinear function depending on a few
variables. The method is based on the inspection of the behavior of discrepan-
cies of a multi-penalty regularization with a component-wise penalization for
small and large values of regularization parameters. We provide the justifica-
tion of the proposed method under a certain condition on sampling operators.
The effectiveness of the method is demonstrated in the example with synthetic
data and in the reconstruction of gene regulatory networks. In the latter ex-
ample, the obtained results provide a clear evidence of the competitiveness of
the proposed method.

The obvious and highly accepted convenience of smartphone apps will, already in the nearest future, bring new opportunities for diabetes therapy management. In particular, it is expected that smartphones will be able to read, store, and display the blood glucose concentration from the continuous glucose monitoring systems. Using our knowledge and experience gained in the framework of the large-scale European Union FP7 funded project ``DIAdvisor: personal glucose predictive diabetes advisor'' (2008-2012), we explore a possibility to develop a novel smartphone app for diabetes patients that provides estimations of the future blood glucose concentration from current and past blood glucose readings. In addition to reliable clinical accuracy, a prediction algorithm implemented in such an app should satisfy multiple requirements, such as easily and quickly implementable on any mobile operating system, portability from individual to individual without readjustment or retraining procedure, and a low battery usage feature. In this study, we present a description of the prediction algorithm, developed in the course of the DIAdvisor project, and its version on Android OS that meets the above-mentioned requirements. Additionally, we compare the clinical accuracy of the algorithm with the state-of-the-art in terms of the ``gold standard'' metric, Clarke Error Grid Analysis, and the recently introduced metric, Prediction-Error Grid Analysis.

Granger causality, based on a vector autoregressive model, is one of the most popular methods for uncovering the temporal dependencies between time series. The application of Granger causality to detect inference among a big number of variables (genes) requires a variable selection procedure. To fight with lack of informative data, the so called regularization procedures are applied. In this chapter, we review current literature applying Granger causality with Lasso regularization technique for ill-posed problems. We discuss regularization procedures for inverse and ill-posed problems and present our recent approaches that are evaluated in a case study of gene regulatory networks reconstruction.

Making accurate predictions is a crucial factor in many systems. The situation encountered in real-life applications is to have only at disposal incomplete/ rough high-dimensional data, and
extracting predictive model from them is an impossible task unless one relies on some a-priori knowledg of properties of expected model. To
overcome these fundamental challenges, we incorporate additional information through optimization by means of multiparameter
regularization. The main goals of the proposed minisymposium are to set up a new agenda and give a new impulse to the cooperation between
approximation and regularization theories within the intrinsic uncertainty of learning process for real-life data.

In this paper we present a new scheme of a kernel-based regularization learning algorithm, in which the kernel and the regularization parameter are adaptively chosen on the base of previous experience with similar learning tasks. The construction of such a scheme is motivated by the problem of prediction of the blood glucose levels of diabetic patients. We describe how the proposed scheme can be used for this problem and report the results of the tests with real clinical data as well as comparing them with existing literature.