Encoding the \ell_p Ball from Limited Measurements
DCC '06 Proceedings of the Data Compression Conference
Subspace pursuit for compressive sensing signal reconstruction
IEEE Transactions on Information Theory
Number of measurements in sparse signal recovery
ISIT'09 Proceedings of the 2009 IEEE international conference on Symposium on Information Theory - Volume 1
Bayesian compressive sensing via belief propagation
IEEE Transactions on Signal Processing
IEEE Transactions on Signal Processing
A statistical-mechanics approach to large-system analysis of CDMA multiuser detectors
IEEE Transactions on Information Theory
Randomly spread CDMA: asymptotics via statistical physics
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
IEEE Transactions on Information Theory
Multiuser Detection of Sparsely Spread CDMA
IEEE Journal on Selected Areas in Communications
Bayesian compressive sensing via belief propagation
IEEE Transactions on Signal Processing
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Compressed sensing deals with the reconstruction of a high-dimensional signal from far fewer linear measurements, where the signal is known to admit a sparse representation in a certain linear space. The asymptotic scaling of the number of measurements needed for reconstruction as the dimension of the signal increases has been studied extensively. This work takes a fundamental perspective on the problem of inferring about individual elements of the sparse signal given the measurements, where the dimensions of the system become increasingly large. Using the replica method, the outcome of inferring about any fixed collection of signal elements is shown to be asymptotically decoupled, i.e., those elements become independent conditioned on the measurements. Furthermore, the problem of inferring about each signal element admits a single-letter characterization in the sense that the posterior distribution of the element, which is a sufficient statistic, becomes asymptotically identical to the posterior of inferring about the same element in scalar Gaussian noise. The result leads to simple characterization of all other elemental metrics of the compressed sensing problem, such as the mean squared error and the error probability for reconstructing the support set of the sparse signal. Finally, the single-letter characterization is rigorously justified in the special case of sparse measurement matrices where belief propagation becomes asymptotically optimal.