Projektdetails
Abstract
Wider research context
Whereas in the last decade the mathematical optimization community focused its efforts on providing algorithms for convex optimization problems, the nonconvex framework has been only timidly approached, the focus being put on methods for solving basic optimization problems. While the usefulness of nonconvex
optimization for signal processing has been recognized recently, the full connection from the mathematical theory to implementation to applications in acoustics is still lacking.
Objectives
The principal aim is to develop novel nonsmooth nonconvex optimization methods and apply them to various problems arising in acoustic signal processing. In the challenging theoretical part, we will design new numerical schemes and provide a rigorous mathematical analysis of their convergence, accuracy and
stability. We will address optimization problems with complex structures in the spirit of the full splitting paradigm, while a particular attention will be paid to the difference-of-convex and fractional optimization problems, for which, despite the fact that they are per se nonconvex, one can exploit the convexity of the
individual components of the objective functions. Furthermore, we will investigate the impact of inertial and memory effects on their convergence. We will apply the resulting algorithms for novel approaches to problems in various acoustic signal fields, in particular to sparsity and compressive sensing, audio
inpainting, system identification and phase retrieval. In the sense of reproducible research, the developed algorithms will be included in an open-source toolbox.
Methods
We plan to carry out a convergence and a convergence rate analysis for the developed numerical algorithms relying on the interplay between continuous and discrete time approaches, Lyapunov-type specific techniques, and variational and nonsmooth analysis concepts and results. We will first investigate the subsequence convergence of the generated iterates to KKT points defined in terms of the limiting subdifferential, and then their global convergence in the framework of the Kurdyka - Lojasiewicz property. 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, that are currently active research topics there.
Level of originality
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.
Primary researchers involved
Radu Ioan Bot (applicant), Peter Balazs (national research partner)
Whereas in the last decade the mathematical optimization community focused its efforts on providing algorithms for convex optimization problems, the nonconvex framework has been only timidly approached, the focus being put on methods for solving basic optimization problems. While the usefulness of nonconvex
optimization for signal processing has been recognized recently, the full connection from the mathematical theory to implementation to applications in acoustics is still lacking.
Objectives
The principal aim is to develop novel nonsmooth nonconvex optimization methods and apply them to various problems arising in acoustic signal processing. In the challenging theoretical part, we will design new numerical schemes and provide a rigorous mathematical analysis of their convergence, accuracy and
stability. We will address optimization problems with complex structures in the spirit of the full splitting paradigm, while a particular attention will be paid to the difference-of-convex and fractional optimization problems, for which, despite the fact that they are per se nonconvex, one can exploit the convexity of the
individual components of the objective functions. Furthermore, we will investigate the impact of inertial and memory effects on their convergence. We will apply the resulting algorithms for novel approaches to problems in various acoustic signal fields, in particular to sparsity and compressive sensing, audio
inpainting, system identification and phase retrieval. In the sense of reproducible research, the developed algorithms will be included in an open-source toolbox.
Methods
We plan to carry out a convergence and a convergence rate analysis for the developed numerical algorithms relying on the interplay between continuous and discrete time approaches, Lyapunov-type specific techniques, and variational and nonsmooth analysis concepts and results. We will first investigate the subsequence convergence of the generated iterates to KKT points defined in terms of the limiting subdifferential, and then their global convergence in the framework of the Kurdyka - Lojasiewicz property. 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, that are currently active research topics there.
Level of originality
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.
Primary researchers involved
Radu Ioan Bot (applicant), Peter Balazs (national research partner)
| Status | Laufend |
|---|---|
| Tatsächlicher Beginn/ -es Ende | 21/06/21 → 20/06/25 |
Projektbeteiligte
- Universität Wien (Leitung)
- Österreichische Akademie der Wissenschaften (ÖAW)