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SPALORA

Sparse and Low Rank Recovery

 

Principal Investigator

Prof. Dr. Holger Rauhut
RWTH Aachen
Lehrstuhl für Mathematik C (Analysis)
Templergraben 55
52056 Aachen

 

Abstract

Compressive sensing is a novel field in signal processing at the interface of applied mathematics, electrical engineering and computer science, which caught significant interest over the past five years. It provides a fundamentally new approach to signal acquisition and processing that has large potential for many applications. It is based on the empirical observation that many signals appearing in real-world applications can be well-approximated by a sparse expansion. Compressive sensing (sparse recovery) predicts the surprising phenomenon that such signals can be recovered from what was previously believed to be highly incomplete measurements (information) using computationally efficient algorithms. In the past year, exciting new developments emerged on the heels of compressive sensing: low rank matrix recovery (matrix completion); as well as a novel approach to the recovery of high-dimensional functions.

We plan to pursue the following research directions:

  • Compressive Sensing: We will investigate several open important mathematical problems, such as the rigorous analysis of certain measurement matrices.
  • Low rank matrix recovery: In low rank matrix recovery, one replaces the sparsity assumption by a lowrank assumption. First results predict that low rank matrices can be recovered from incomplete linearinformation using convex optimization.
  • Low rank tensor recovery: We plan to extend methods and mathematical results from low rank matrix recovery to tensors. This field is presently completely open.
  • Recovery of high-dimensional functions: Classical methods for the numerical treatment of highdimensionalfunctions commonly suffer from the curse of dimensionality: the computational effort increasesdramatically with growing dimension. In order to decrease the computational burden, a recentnovel approach assumes that the function of interest actually depends only on a small number of a priori unknown variables. Preliminary results suggest that compressive sensing and low rank matrix recovery tools can be applied to the efficient recovery of such functions.

We plan to develop computational methods for all the subtopics and to derive rigorous mathematical results on their performance. With the experience I gained over the past years, I strongly believe that I have the necessary competence to pursue this project. I expect a strong impact in science and technology.

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