An Estimation Theoretic Approach for Sparsity Pattern Recovery in the Noisy Setting

Compressed sensing deals with the reconstruction of sparse signals using a small number of linear measurements. One of the main challenges in compressed sensing is to find the support of a sparse signal. In the literature, several bounds on the scaling law of the number of measurements for successful support recovery have been derived where the main focus is on random Gaussian measurement matrices. In this paper, we investigate the noisy support recovery problem from an estimation theoretic point of view, where no specific assumption is made on the underlying measurement matrix. The linear measurements are perturbed by additive white Gaussian noise. We define the output of a support estimator to be a set of position values in increasing order. We set the error between the true and estimated supports as the l2-norm of their difference. On the one hand, this choice allows us to use the machinery behind the l2-norm error metric and on the other hand, converts the support recovery into a more intuitive and geometrical problem. First, by using the Hammersley-Chapman-Robbins (HCR) bound, we derive a fundamental lower bound on the performance of any unbiased estimator of the support set. This lower bound provides us with necessary conditions on the number of measurements for reliable l2-norm support recovery, which we specifically evaluate for uniform Gaussian measurement matrices. Then, we analyze the maximum likelihood estimator and derive conditions under which the HCR bound is achievable. This leads us to the number of measurements for the optimum decoder which is sufficient for reliable l2-norm support recovery and shows that the performance of the optimum decoder has only a 9 dB gap The authors are with the School of Computer and Communication Sciences, Ecole Polytechnique Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Switzerland (e-mails: {ali.hormati, amin.karbasi, soheil.mohajer, martin.vetterli}@epfl.ch). Martin Vetterli is also with the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA 94720, USA. This work was supported by the Swiss National Science Foundation under grants NCCR-MICS-51NF40-111400 and 200020103729. The material in this work was presented in part at the IEEE International Symposium on Information Theory, Seoul, Korea, June 2009. October 28, 2013 DRAFT