Beyond convergence rates: Exact recovery with Tikhonov regularization with sparsity constraints

The Tikhonov regularization of linear ill-posed problems with an $\ell^1$ penalty is considered. We recall results for linear convergence rates and results on exact recovery of the support. Moreover, we derive conditions for exact support recovery which are especially applicable in the case of ill-posed problems, where other conditions, e.g. based on the so-called coherence or the restricted isometry property are usually not applicable. The obtained results also show that the regularized solutions do not only converge in the $\ell^1$-norm but also in the vector space $\ell^0$ (when considered as the strict inductive limit of the spaces $\R^n$ as $n$ tends to infinity). Additionally, the relations between different conditions for exact support recovery and linear convergence rates are investigated. With an imaging example from digital holography the applicability of the obtained results is illustrated, i.e. that one may check a priori if the experimental setup guarantees exact recovery with Tikhonov regularization with sparsity constraints

Bridging the gap between systems biology and medicine

Systems biology has matured considerably as a discipline over the last decade, yet some of the key challenges separating current research efforts in systems biology and clinically useful results are only now becoming apparent. As these gaps are better defined, the new discipline of systems medicine is emerging as a translational extension of systems biology. How is systems medicine defined? What are relevant ontologies for systems medicine? What are the key theoretic and methodologic challenges facing computational disease modeling? How are inaccurate and incomplete data, and uncertain biologic knowledge best synthesized in useful computational models? Does network analysis provide clinically useful insight? We discuss the outstanding difficulties in translating a rapidly growing body of data into knowledge usable at the bedside. Although core-specific challenges are best met by specialized groups, it appears fundamental that such efforts should be guided by a roadmap for systems medicine drafted by a coalition of scientists from the clinical, experimental, computational, and theoretic domains