Fundamentals of statistical signal processing: estimation theory
Fundamentals of statistical signal processing: estimation theory
Sparse Approximate Solutions to Linear Systems
SIAM Journal on Computing
Atomic Decomposition by Basis Pursuit
SIAM Journal on Scientific Computing
Sparse bayesian learning and the relevance vector machine
The Journal of Machine Learning Research
Information and Complexity in Statistical Modeling
Information and Complexity in Statistical Modeling
Compressive data gathering for large-scale wireless sensor networks
Proceedings of the 15th annual international conference on Mobile computing and networking
Subspace pursuit for compressive sensing signal reconstruction
IEEE Transactions on Information Theory
A sparse signal reconstruction perspective for source localization with sensor arrays
IEEE Transactions on Signal Processing - Part II
Sparse Bayesian learning for basis selection
IEEE Transactions on Signal Processing
An affine scaling methodology for best basis selection
IEEE Transactions on Signal Processing
Sparse signal reconstruction from limited data using FOCUSS: are-weighted minimum norm algorithm
IEEE Transactions on Signal Processing
IEEE Transactions on Signal Processing
Decoding by linear programming
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit
IEEE Transactions on Information Theory
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We propose a variance-component probabilistic model for sparse signal reconstruction and model selection. The measurements follow an underdetermined linear model, where the unknown regression vector (signal) is sparse or approximately sparse and noise covariance matrix is known up to a constant. The signal is composed of two disjoint parts: a part with significant signal elements and the complementary part with insignificant signal elements that have zero or small values. We assign distinct variance components to the candidates for the significant signal elements and a single variance component to the rest of the signal; consequently, the dimension of our model's parameter space is proportional to the assumed sparsity level of the signal. We derive a generalized maximum-likelihood (GML) rule for selecting the most efficient parameter assignment and signal representation that strikes a balance between the accuracy of data fit and compactness of the parameterization. We prove that, under mild conditions, the GML-optimal index set of the distinct variance components coincides with the support set of the sparsest solution to the underlying underdetermined linear system. Finally, we propose an expansion-compression variance-component based method (ExCoV) that aims at maximizing the GML objective function and provides an approximate GML estimate of the significant signal element set and an empirical Bayesian signal estimate. The ExCoV method is automatic and demands no prior knowledge about signal-sparsity or measurement-noise levels. We also develop a computationally and memory efficient approximate ExCoV scheme suitable for large-scale problems, apply the proposed methods to reconstruct one- and two-dimensional signals from compressive samples, and demonstrate their reconstruction performance via numerical simulations. Compared with the competing approaches, our schemes perform particularly well in challenging scenarios where the noise is large or the number of measurements is small.