Reconstruction of signals drawn from a gaussian mixture via noisy compressive measurements


Journal Article

This paper determines to within a single measurement the minimum number of measurements required to successfully reconstruct a signal drawn from a Gaussian mixture model in the low-noise regime. The method is to develop upper and lower bounds that are a function of the maximum dimension of the linear subspaces spanned by the Gaussian mixture components. The method not only reveals the existence or absence of a minimum mean-squared error (MMSE) error floor (phase transition) but also provides insight into the MMSE decay via multivariate generalizations of the MMSE dimension and the MMSE power offset, which are a function of the interaction between the geometrical properties of the kernel and the Gaussian mixture. These results apply not only to standard linear random Gaussian measurements but also to linear kernels that minimize the MMSE. It is shown that optimal kernels do not change the number of measurements associated with the MMSE phase transition, rather they affect the sensed power required to achieve a target MMSE in the low-noise regime. Overall, our bounds are tighter and sharper than standard bounds on the minimum number of measurements needed to recover sparse signals associated with a union of subspaces model, as they are not asymptotic in the signal dimension or signal sparsity. © 2014 IEEE.

Full Text

Duke Authors

Cited Authors

  • Renna, F; Calderbank, R; Carin, L; Rodrigues, MRD

Published Date

  • May 1, 2014

Published In

Volume / Issue

  • 62 / 9

Start / End Page

  • 2265 - 2277

International Standard Serial Number (ISSN)

  • 1053-587X

Digital Object Identifier (DOI)

  • 10.1109/TSP.2014.2309560

Citation Source

  • Scopus