Title: "Compressed Sensing: what it is, key results and recent trends"

Abstract:

The well-known theory of Compressed Sensing explores the recovery of vectors x \in R^N from incomplete linear information y = Ax \in R^m , where A \in R^{m×N}, m << N, and R is the set of real numbers. To make this possible, Compressed Sensing relies on two principles: sparsity, which pertains to the signals of interest, and incoherence, which pertains to the sensing modality. The current presentation is going to explore key-results and solving strategies used in the Compressed Sensing problem. First, the sparse representation synthesis model and analysis sparsity model (also known as co-sparse model) will be presented, which look at the compressed sensing problem from different viewpoints. Then, a tour around the algorithms used in Compressed Sensing will be made, along with conditions on the matrix A which ensure exact or approximate reconstruction of the original sparse vector x. Furthermore, another interesting question is the minimal number of linear measurements needed to reconstruct sparse vectors from these measurements, where notions as the compressive widths appear (quantities of this type play a crucial role in the modern field of information based complexity).

Apart from the theoretical framework developed above, there is a wide variety of scientific areas where the theory of Compressed Sensing is applied; Magnetic Resonance Imaging (MRI), Security and Cryptography, Radar and Telecommunications, Sampling Theory, Computer Graphics, Statistics and Machine Learning, Deep Learning, just to name a few. Therefore, it would be very insightful to see how problems from some of the aforementioned fields can be modeled as standard Compressed Sensing problems and of course, recent advances in them will be presented.

Bio:

Vicky Kouni obtained both her BSc (Ptychion) and her MSc in Applied Mathematics from the Department of Mathematics of National and Kapodistrian University of Athens. She is presently a PhD student at the Department of Informatics and Telecommunications of National and Kapodistrian University of Athens, under the supervision of Prof. Theoharis Theoharis. She is also co-advised by Prof. Holger Rauhut from RWTH Aachen University. Her research interests lie on Compressed Sensing, Sparse Representantions, Applied Harmonic Analysis and their applications, especially in the domain of Imaging Sciences.