Variational methods are widely applied to ill-posed inverse problems for they have the ability to embed prior knowledge about the solution. However, the level of performance of these methods significantly depends on a set of parameters, which can be estimated through computationally expensive and time-consuming methods. In contrast, deep learning offers very generic and efficient architectures, at the expense of explainability, since it is often used as a black-box, without any fine control over its output. Deep unfolding provides a convenient approach to combine variational-based and deep learning approaches. Starting from a variational formulation for image restoration, we develop iRestNet, a neural network architecture obtained by unfolding a proximal interior point algorithm. Hard constraints, encoding desirable properties for the restored image, are incorporated into the network thanks to a logarithmic barrier, while the barrier parameter, the stepsize, and the penalization weight are learned by the network. We derive explicit expressions for the gradient of the proximity operator for various choices of constraints, which allows training iRestNet with gradient descent and backpropagation. In addition, we provide theoretical results regarding the stability of the network for a common inverse problem example. Numerical experiments on image deblurring problems show that the proposed approach compares favorably with both state-of-the-art variational and machine learning methods in terms of image quality.

The goal of this article is to promote the use of fixed point strategies in data science by showing that they provide a simplifying and unifying framework to model, analyze, and solve a great variety of problems. They are seen to constitute a natural environment to explain the behavior of advanced convex optimization methods as well as of recent nonlinear methods in data science which are formulated in terms of paradigms that go beyond minimization concepts and involve constructs such as Nash equilibria or monotone inclusions. We review the pertinent tools of fixed point theory and describe the main state-of-the-art algorithms for provenly convergent fixed point construction. We also incorporate additional ingredients such as stochasticity, block-implementations, and non-Euclidean metrics, which provide further enhancements. Applications to signal and image processing, machine learning, statistics, neural networks, and inverse problems are discussed.

The dimension and the complexity of inference problems have dramatically increased in statistical signal processing. It thus becomes mandatory to design improved proposal schemes in Metropolis-Hastings algorithms, providing large proposal transitions that are accepted with high probability. The proposal density should ideally provide an accurate approximation of the target density with a low computational cost. In this paper, we derive a novel Metropolis-Hastings proposal, inspired from Langevin dynamics, where the drift term is preconditioned by an adaptive matrix constructed through a Majorization-Minimization strategy. We propose several variants of low-complexity curvature metrics applicable to large scale problems. We demonstrate the geometric ergodicity of the resulting chain for the class of super-exponential distributions. The proposed method is shown to exhibit a good performance in two signal recovery examples.

Underdetermined or ill-posed inverse problems require additional information for sound solutions with tractable optimization algorithms. Sparsity yields consequent heuristics to that matter, with numerous applications in signal restoration, image recovery, or machine learning. Since the $\ell_0$ count measure is barely tractable, many statistical or learning approaches have invested in computable proxies, such as the $\ell$ _1$ norm. However, the latter does not exhibit the desirable property of scale invariance for sparse data. Extending the SOOT Euclidean/Taxicab $\ell_1$-over-$\ell_2$ norm-ratio initially introduced for blind deconvolution, we propose SPOQ, a family of smoothed (approximately) scale-invariant penalty functions. It consists of a Lipschitz-differentiable surrogate for $\ell_p$-over-$\ell_q$ quasi-norm/norm ratios with $p\in\,]0,2[$ and $q\ge 2$ . This surrogate is embedded into a novel majorize-minimize trust-region approach, generalizing the variable metric forward-backward algorithm. For naturally sparse mass-spectrometry signals, we show that SPOQ significantly outperforms $\ell_0$ , $\ell_1$ , Cauchy, Welsch, SCAD and Celo penalties on several performance measures. Guidelines on SPOQ hyperparameters tuning are also provided, suggesting simple data-driven choices.