Unlabelled Sensing: A Sparse Bayesian Learning Approach

We address the recovery of sparse vectors in an overcomplete, linear and noisy multiple measurement framework, where the measurement matrix is known upto a permutation of its rows. We derive sparse Bayesian learning (SBL) based updates for joint recovery of the unknown sparse vectors and the sensing order, represented using a permutation matrix. We model the sparse vectors using multiple uncorrelated and correlated vectors, and in particular, we use the first order AR model for the correlated sparse vectors. We propose the Permutation-MSBL and a Kalman filtering based PermutationKSBL algorithm for low-complexity joint recovery of the sparse vectors and the permutation matrix. The novelty of this work is in providing a simple update step for the permutation matrix using the rearrangement inequality. We demonstrate the mean square error and the permutation recovery performance of the proposed algorithms as compared to a compressed sensing based scheme. EDICS: SAS-STAT, MLSAS-SPARSE, MLSAS-BAYES, SAS-ADAP