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Sparse and Low Rank Recovery

Final Report Summary - SPALORA (Sparse and Low Rank Recovery)

The ability to efficiently process data has become increasingly important in our present digital age. The recent theory of compressive sensing (or sparse recovery) investigates ways of acquiring signals (audio, images, video etc.) from what was previously considered incomplete information. The idea is to exploit the observation that many types of signals are compressible well, which suggests that much fewer information is necessary than the actual signal length in order to reconstruct the signal from measured data. It is, however, nontrivial to come up with suitable measurement schemes and efficient reconstruction algorithms. The theory connects to several mathematical areas such as random matrices, convex optimization and approximation theory. Many applications can profit from this new mathematical signal processing methodology as measurement time and/or costs can be significantly reduced, or the quality of reconstructed signals/images improves. Examples of such applications include magnetic resonance imaging, radar, astronomy, wireless communications and more.

This project investigates the mathematics of specific measurement processes arising for instance in radar imaging and develops far reaching generalizations and applications of the theory.

As a key result, the project established a significantly improved estimates of the number of measurements required for reconstruction in various contexts. A particular one of fundamental importance considers convolutions with a random vector, a measurement scenario that arises for instance in radar and optical imaging. The basic mathematical tool developed for this purpose is likely to be very useful also in other contexts. Further results of this type were achieved for two classes of radar systems: MIMO radar with random waveforms and arrays with randomly positioned antennas. Compressive sensing techniques turned out to be particularly promising in radar.

Compressibility is usually referred to as the ability to approximate a signal well by an expansion having only few nonzero coefficients in a suitable basis (this is for instance the concept behind the JPEG image compression standard). A generalization considers the approximation of a higher-dimensional array - a so called tensor, or a matrix in the two-dimensional case – by tensors of low rank. Again low-rank tensors can be described by few parameters, but in contrast to standard compressibility a corresponding basis is not fixed beforehand so that this concept is much more flexible. The problem of recovering low rank tensors from incomplete information turned out to be mathematically very difficult. Nevertheless, the project developed new recovery algorithms which empirically work very well. At least, a partial theoretical analysis was achieved.

A quantum state can be modeled with a low rank matrix, and it is a challenge to determine a state from measurements. The project could derive theoretical guarantees on the recovery of quantum state from a minimal number of certain measurements. For the first time, the theoretical results can also treat measurement systems that can (at least in principle) be realized in actual quantum physical experiments and may be an ingredient in future quantum computers.

The interpolation of functions from sample values has many important applications in signal processing and numerical analysis. Based on concepts from compressive sensing, the project developed new methods together with their theoretical analysis which improve over classical methods in particular for functions of many variables. These methods are expected to have impact also for the numerical solution of partial differential equations with uncertain parameters (uncertainty quantification), which is relevant for simulation of real-world problems where model parameters often cannot be determined exactly.

Moreover, a book on the mathematical foundations of compressive sensing was finalized which serves as a basic introduction to this new field and contains a number of new result achieved within the project. It is the first monograph on this subject and already has significant impact on the research and teaching in the area.

It can be expected that the developed methods and underlying results of this project will end up in technological products/standards for instance, in the areas of wireless communications, radar, quantum computers, medical imaging, simulation technology and more.