In recent years, non-linear optimization has had a crucial role in the development of modern techniques at the interface of machine learning and imaging. The present book is a collection of recent contributions in the field of optimization, either revisiting consolidated ideas to provide formal theoretical guarantees or providing comparative numerical studies for challenging inverse problems in imaging. The work of these papers originated in the INdAM Workshop “Advanced Techniques in Optimization for Machine learning and Imaging” held in Roma, Italy, on June 20-24, 2022. The covered topics include non-smooth optimisation techniques for model-driven variational regularization, fixed-point continuation algorithms and their theoretical analysis for selection strategies of the regularization parameter for linear inverse problems in imaging, different perspectives on Support Vector Machines trained via Majorization-Minimization methods, generalization of Bayesian statistical frameworks to imaging problems, and creation of benchmark datasets for testing new methods and algorithms. Preface Contents About the Editors STEMPO—Dynamic X-Ray Tomography Phantom 1 Introduction 2 Phantom Structure 3 Data Set Contents 3.1 Sinograms 3.2 Complementary Data 3.3 Forward Operators 4 Measurements 5 Example Codes 6 Future Development References On a Fixed-Point Continuation Method for a Convex Optimization Problem 1 Introduction 2 Trade-Off Curve 3 Convergence Analysis 4 Numerical Experiments 4.1 Approximating the Trade-Off Curve 5 Conclusions References Majoration-Minimization for Sparse SVMs 1 Introduction 2 Problem Formulation 2.1 SVM Loss Function 2.2 Regularization 2.3 General Formulation 3 Majorization Properties 3.1 Descent Lemma Majorant 3.2 Half-Quadratic Majorant 4 Training SVMs 4.1 Gradient Descent Approach 4.2 MM Quadratic Approach 4.3 Stochastic Minimization Approaches 5 Numerical Experiments 5.1 Results 6 Conclusions References Bilevel Learning of Regularization Models and Their Discretization for Image Deblurring and Super-Resolution 1 Introduction 2 Bilevel Learning of FoE Regularization 3 Bilevel Learning of TV Discretization 4 Numerical Experiments 4.1 FoE Bilevel Learning for Image Deblurring 4.2 TV Discretization Learning for Image Deblurring and Super-Resolution 5 Conclusions References Non-Log-Concave and Nonsmooth Sampling via Langevin Monte Carlo Algorithms 1 Introduction 1.1 Related Work 1.2 Notation 2 Preliminaries 2.1 Problem Formulation 2.2 Convex Analysis 3 Langevin Monte Carlo Algorithms 3.1 Unadjusted Langevin Algorithm 3.2 Metropolis-Adjusted Langevin Algorithm 3.3 Preconditioned Unadjusted Langevin Algorithm 3.4 Mirror-Langevin Algorithm 3.5 Comparison of Langevin Monte Carlo Algorithms 4 Proximal Langevin Monte Carlo Algorithms 4.1 Moreau–Yosida Unadjusted Langevin Algorithm 4.2 Proximal Gradient Langevin Dynamics 4.3 Proximal MALA and Moreau–Yosida Regularized MALA 4.4 Preconditioned Proximal Unadjusted Langevin Algorithm 4.5 Forward-Backward Unadjusted Langevin Algorithm 4.6 Bregman–Moreau Unadjusted Mirror-Langevin Algorithm 4.7 Primal-Dual Langevin Algorithms 4.8 Comparison of Proximal Langevin Monte Carlo Algorithms 5 Stochastic Gradient Langevin Monte Carlo Algorithms 6 Numerical Simulations 6.1 Mixtures of Gaussians 6.2 Mixtures of Laplacians 6.3 Mixtures of Gaussians with Laplacian Priors 6.4 Bayesian Image Deconvolution 7 Conclusion References On the Inexact Proximal Gauss–Newton Methods for Regularized Nonlinear Least Squares Problems 1 Introduction 2 The Algorithm and Its Convergence Analysis 3 Numerical Experiments 3.1 Convex Problems 3.2 Non-convex Problems 4 Auxiliary Results References
