Nonsmooth Nonconvex Optimization Methods in Acoustics

The most important problems of current interest in the field of acoustic signal processing include the following: compressed sensing a technique that aims to reconstruct a signal from only a few measurements; audio denoise removing noise or other disturbances from a signal; audio inpainting restoring and recovering missing portions in audio signals; system identification estimating the transformation system from the output signal; and phase retrieval to recover a signal from its magnitudes only. These problems share the feature that they can be modelled and formulated as structured nonsmooth nonconvex optimization problems, which means that, in order to solve them, one usually has to minimize a function which in many cases has a complicated expression and it is neither convex nor differentiable. In other words, the function to be minimized has usually not a global minimum, but many local minima and maxima, and it fails to be differentiable, in particular at these local extrema. In practice many approaches apply ad-hoc, smooth or convex methods, ignoring that they are certainly not perfectly fitted, but still reaching successful solutions. In this project we aim at a more holistic approach marrying mathematics and applications. The main aim of this research project is to design numerical algorithms for solving such structured nonsmooth nonconvex optimization problems without heuristic simplifications. The proposed algorithms will be analyzed from the point of view of their convergence properties, accuracy and stability. The applications to audio signal processing problems will help to validate the theoretically founded convergence behavior of the new algorithms, and also provide new understanding and novel approaches for important tasks in acoustics, like the ones mentioned above. The theoretical goals are at the cutting edge of current mathematical research, therefore their application in signal processing will be extremely innovative. The goal of this application-oriented mathematics project, is not only to apply completely novel mathematical results to certain tasks, but also learn from those applications new concepts and properties that are interesting from a purely mathematical point of view.