Low Complexity Regularization of Linear Inverse Problems

Inverse problems and regularization theory is a central theme in contemporary signal processing, where the goal is to reconstruct an unknown signal from partial indirect, and possibly noisy, measurements of it. A now standard method for recovering the unknown signal is to solve a convex optimization problem that enforces some prior knowledge about its structure. This has proved efficient in many problems routinely encountered in imaging sciences, statistics and machine learning. This chapter delivers a review of recent advances in the field where the regularization prior promotes solutions conforming to some notion of simplicity/low-complexity. These priors encompass as popular examples sparsity and group sparsity (to capture the compressibility of natural signals and images), total variation and analysis sparsity (to promote piecewise regularity), and low-rank (as natural extension of sparsity to matrix-valued data). Our aim is to provide a unified treatment of all these regularizations under a single umbrella, namely the theory of partial smoothness. This framework is very general and accommodates all low-complexity regularizers just mentioned, as well as many others. Partial smoothness turns out to be the canonical way to encode low-dimensional models that can be linear spaces or more general smooth manifolds. This review is intended to serve as a one stop shop toward the understanding of the theoretical properties of the so-regularized solutions. It covers a large spectrum including: (i) recovery guarantees and stability to noise, both in terms of $\ell^2$-stability and model (manifold) identification; (ii) sensitivity analysis to perturbations of the parameters involved (in particular the observations), with applications to unbiased risk estimation ; (iii) convergence properties of the forward-backward proximal splitting scheme, that is particularly well suited to solve the corresponding large-scale regularized optimization problem

Prizes for basic research: Human capital, economic might and the shadow of history

Global economic trends, Basic research, Human capital, Winner-takes-all, F15, F21, O3, N4,

Chronic Akt1 deficiency attenuates adverse remodeling and enhances angiogenesis after myocardial infarction

Background Akt1 is a key signaling molecule in multiple cell types, including endothelial cells. Accordingly, Akt1 was proposed as a therapeutic target for ischemic injury in the context of myocardial infarction (MI). The aim of this study was to use multimodal in vivo imaging to investigate the impact of systemic Akt1 deficiency on cardiac function and angiogenesis before and after MI. Methods and Results In vivo cardiac MRI was performed before and at days 1, 8, 15, and 29 to 30 after MI induction for wild-type, heterozygous, and Akt1-deficient mice. Noninfarcted hearts were imaged using ex vivo stereomicroscopy and microcomputed tomography. Histological examination was performed for noninfarcted hearts and for hearts at days 8 and 29 to 30 after MI. MRI revealed mildly decreased baseline cardiac function in Akt1 null mice, whereas ex vivo stereomicroscopy and microcomputed tomography revealed substantially reduced coronary macrovasculature. After MI, Akt1(-/-) mice demonstrated significantly attenuated ventricular remodeling and a smaller decrease in ejection fraction. At 8 days after MI, a larger functional capillary network at the remote and border zone, accompanied by reduced scar extension, preserved cardiac function, and enhanced border zone wall thickening, was observed in Akt1(-/-) mice when compared with littermate controls. Conclusions Using multimodal imaging to probe the role of Akt1 in cardiac function and remodeling after MI, this study revealed reduced adverse remodeling in Akt1-deficient mice after MI. Augmented myocardial angiogenesis coupled with a more functional myocardial capillary network may facilitate revascularization and therefore be responsible for preservation of infarcted myocardium