Explainable bilevel optimization: an application to the Helsinki deblur challenge

In this paper we present a bilevel optimization scheme for the solution of a general image deblurring problem, in which a parametric variational-like approach is encapsulated within a machine learning scheme to provide a high quality reconstructed image with automatically learned parameters. The ingredients of the variational lower level and the machine learning upper one are specifically chosen for the Helsinki Deblur Challenge 2021, in which sequences of letters are asked to be recovered from out-of-focus photographs with increasing levels of blur. Our proposed procedure for the reconstructed image consists in a fixed number of FISTA iterations applied to the minimization of an edge preserving and binarization enforcing regularized least-squares functional. The parameters defining the variational model and the optimization steps, which, unlike most deep learning approaches, all have a precise and interpretable meaning, are learned via either a similarity index or a support vector machine strategy. Numerical experiments on the test images provided by the challenge authors show significant gains with respect to a standard variational approach and performances comparable with those of some of the proposed deep learning based algorithms which require the optimization of millions of parameters.

A scaled gradient projection method for Bayesian learning in dynamical systems

A crucial task in system identification problems is the selection of the most appropriate model class, and is classically addressed resorting to cross-validation or using asymptotic arguments. As recently suggested in the literature, this can be addressed in a Bayesian framework, where model complexity is regulated by few hyperparameters, which can be estimated via marginal likelihood maximization. It is thus of primary importance to design effective optimization methods to solve the corresponding optimization problem. If the unknown impulse response is modeled as a Gaussian process with a suitable kernel, the maximization of the marginal likelihood leads to a challenging nonconvex optimization problem, which requires a stable and effective solution strategy. In this paper we address this problem by means of a scaled gradient projection algorithm, in which the scaling matrix and the steplength parameter play a crucial role to provide a meaning solution in a computational time comparable with second order methods. In particular, we propose both a generalization of the split gradient approach to design the scaling matrix in the presence of box constraints, and an effective implementation of the gradient and objective function. The extensive numerical experiments carried out on several test problems show that our method is very effective in providing in few tenths of a second solutions of the problems with accuracy comparable with state-of-the-art approaches. Moreover, the flexibility of the proposed strategy makes it easily adaptable to a wider range of problems arising in different areas of machine learning, signal processing and system identification.