Random Matrix Methods in Compressed Sensing
Speaker: Antonia Tulino
Faculty Host: Professor Elza Erkip
Abstract
Consider a Bernoulli-Gaussian complex n-vector whose components are independent and identically distributed. This random q-sparse vector is multiplied by a random matrix U, and a randomly chosen subset of the components of average size np, 0<p<1, of the resulting vector is then observed in additive Gaussian noise. We extend the scope of conventional noisy compressive sampling models where U is typically the identity or a matrix with iid components, to allow U that
satisfies a certain freeness condition, which encompasses Haar matrices and other unitarily invariant matrices. We use the replica method and the decoupling principle of Guo and Verdu, as well as a number of information theoretic bounds, to study the input-output mutual information and the support recovery error rate as n goes to infinity.
However, most of sparse signal of practical interest have correlated components. As a practical example of sparse correlated signals, we consider the impulse response of a wireless channel and we study the performance of CS in wireless channel estimation comparing it to classical estimation techniques. Focusing on a MIMO wireless channel, we consider a dual perspective on the problem, where both the mean square error distortion achievable in estimating the channel and rate achievable given the estimate of channel, are considered as figures of merit. Motivated by this application, we propose a new algorithm which exploits the available a priori knowledge with high flexibility. To conclude, we present some work in progress concerning information theoretic bounds obtained by using replica methods for CS when sources with correlation are considered.
About the Speaker
Antonia M. Tulino received the Ph.D. degree from the Electrical Engineering Department, Seconda Universitá degli Studi di Napoli, Italy, in 1999. She has served as Associate Professor at the Department of Electrical and Telecommunications Engineering at the Universitá degli Studi di Napoli "Federico II" since 2002. She is currently with the Department of Wireless Communications, Bell Laboratories, Alcatel-Lucent, Holmdel, NJ. She held research positions at the Center for Wireless Communications, Oulu, Finland and at the Department of Electrical Engineering, Princeton University, Princeton, NJ. She has served on the Faculty of Engineering, Universitá degli Studi del Sannio, Benevento, Italy. Dr. Tulino has received the 2009 Stephen O. Rice Prize in the Field of Communications Theory for the best paper published in the IEEE TRANSACTION ON COMMUNICATIONS in 2008. A frequent contributor to the IEEE TRANSACTIONS ON INFORMATION THEORY, the IEEE TRANSACTIONS ON COMMUNICATIONS, and the IEEE TRANSACTIONS ON SIGNAL PROCESSING, her research interests lay in the broad area of communication systems approached with the complementary tools provided by signal processing, information theory, and random matrix theory.