The change of paradigm from purely model driven to data driven (learning based) approaches has tremendously altered the picture in many applications in Machine Learning, Computer Vision, Signal Processing, Inverse Problems, Statistics and so on. There is no need to mention the significant boost in performance for many specific applications, thanks to the advent of large scale Deep Learning. In this talk, we open the area of optimization algorithms for this data driven paradigm, for which theoretical guarantees are indispensable. The expectations about an optimization algorithm are clearly beyond empirical evidence, as there may be a whole processing pipeline depending on a reliable output of the optimization algorithm, and application domains of algorithms can vary significantly. While there is already a vast literature on "learning to optimize", there is no theoretical guarantees associated with these algorithms that meet these expectations from an optimization point of view. We develop the first framework to learn optimization algorithms with provable generalization guarantees to certain classes of optimization problems, while the learning based backbone enables the algorithms' functioning far beyond the limitations of classical (deterministic) worst case bounds. Our results rely on PAC-Bayes bounds for general, unbounded loss-functions based on exponential families. We learn optimization algorithms with provable generalization guarantees (PAC-bounds) and explicit trade-off between a high probability of convergence and a high convergence speed.

The recovery of a signal from the magnitudes of its transformation, like the Fourier transform, is known as the phase retrieval problem and is of big relevance in various fields of engineering and applied physics. The Griffin-Lim algorithm is a staple method commonly used for the phase retrieval problem, which is based on alternating projections. In this talk, we introduce and motivate a fast inertial/momentum modification of the Griffin-Lim Algorithm for the phase retrieval problem and we present a convergence guarantee for the new algorithm.

The goal of audio inpainting is to fill missing data, i.e., gaps, in an acoustical signal. Depending on the length of the gap this procedure should either recreate the original signal, or at least provide a perceptually pleasant and meaningful solution.

We give an overview of existing methods for different gap lengths, and discuss details of our own method [1] for gaps of medium duration. This approach is based on promoting sparsity in the time-frequency domain, combined with a convexifaction using ADMM with a dictionary learning technique that perturbs the time-frequency atoms around the gap, using an optimization technique originally developed in the context of channel estimation.

[1] G. Tauböck, S. Rajbamshi, P. Balazs. Dictionary learning for sparse audio inpainting, IEEE Journal of Selected Topics in Signal Processing 15 no. 1: 104–119, 2021.

New materials like fibre metal laminates (FML) call for new methods when it comes to structural health monitoring (SHM). In this talk we describe an approach to SHM in FML based on guided ultrasonic waves that travel through plates as lamb waves. By the controlled emission of such waves and the measurement of the displacement at a few position, we aim to detect if a damage in the material is present. We approach this inverse problem by an analytical model of the forward propoagation and a simple damage model that is (nonlinearly) parameterized by a small number of parameters. To identify the damage parameters we employ Bayesian methods (namely a Markov Chain Monte-Carlo Metropolis-Hastings method and the ensemble Kalman filter). To make these computationally tractable, we use parametric model reduction to speed up the forward evaluations of the model.