Support Recovery With Sparsely Sampled Free Random Matrices

Abstract

Consider a Bernoulli-Gaussian complex n-vector whose components are V i = X i B i , with X i ∼ CN (0, P x ) and binary B i mutually independent and iid across i. This random q-sparse vector is multiplied by a square random matrix U, and a randomly chosen subset, of average size np, p ∈ [0, 1], of the resulting vector components is then observed in additive Gaussian noise. We extend the scope of conventional noisy compressive sampling models where U is typically a matrix with iid components, to allow U satisfying a certain freeness condition. This class of matrices encompasses Haar matrices and other unitarily invariant matrices. We use the replica method and the decoupling principle of Guo and Verdú, as well as a number of information theoretic bounds, to study the input-output mutual information and the support recovery error rate in the limit of n → ∞. We also extend the scope of the large deviation approach of Rangan, Fletcher and Goyal and characterize the performance of a class of estimators encompassing thresholded linear MMSE and 1 relaxation.

Transmission Strategies for Remote Estimation With an Energy Harvesting Sensor

“…the non-zero eigenvalues are all equal. This type of models have been used to represent signal families with a low degree of freedom in various signal applications, for instance as a sparse signal model in the compressive sensing literature [13], [39]. We obtain the following result for ε LB : Lemma 3.5: Let Λ x,s = (P x /s)I s , H = I n .…”

Improved Asymptotic Capacity Lower Bound for OFDM System with Compressed Sensing Channel Estimation for Bernoulli Gaussian Channel

“…For the model in (2) the asymptotic behaviour (L → ∞) for different estimators can be predicted using the Replica method [14]. In [13] the authors predicted the MMSE error estimate of a reconstruction algorithm obtained by augmenting the Lasso estimator with an MMSE estimator.…”

“…These issues for the case of Lasso based channel estimation in the asymptotic scenario were addressed in a recent paper [13]. In [13], the authors made a formal connection between the pilot aided OFDM channel estimation problem for a Bernoulli-Gaussian channel and the compressed sensing problem with partial DFT sensing matrix for a Bernoulli-Gaussian unknown vector [14]. To avoid correlation between the channel estimate and the estimation error, in [13] a new hybrid Lasso-MMSE estimator was proposed.…”

“…To avoid correlation between the channel estimate and the estimation error, in [13] a new hybrid Lasso-MMSE estimator was proposed. For this estimator the MMSE was found using known results from the Replica method [14], [15] and the asymptotic capacity lower bound was derived based on [1], [16]. Additionally, using the asymptotic capacity lower bound, the authors in [13] proposed an optimization problem for finding the optimal average fraction of pilot subcarriers used for channel estimation and optimal pilot to data power ratio given the average symbol power per subcarrier, which was solved using grid based exhaustive search.…”

“…Here we improve the asymptotic capacity lower bound from [13] by replacing the Lasso based Lasso-MMSE by pure MMSE estimator. The MMSE estimation error of the MMSE estimator can also be predicted using the Replica method [14] and can be achieved for broad range of system parameters using the Turbo compressed sensing recovery algorithm [17], [18]. The improved asymptotic capacity lower bound sheds light on the expected improvement from the utilization of optimal reconstruction algorithms in compressed sensing compared to the widely used Lasso based solutions.…”