Expectation-Maximization Gaussian-Mixture Approximate Message Passing

When recovering a sparse signal from noisy compressive linear measurements, the distribution of the signal's non-zero coefficients can have a profound effect on recovery mean-squared error (MSE). If this distribution was a priori known, then one could use computationally efficient approximate m...

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Bibliographic Details
Published in:IEEE transactions on signal processing Vol. 61; no. 19; pp. 4658 - 4672
Main Authors: Vila, Jeremy P., Schniter, Philip
Format: Journal Article
Language:English
Published: New York, NY IEEE 01.10.2013
Institute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subjects:
Algorithms
Applied sciences
Approximation
Approximation algorithms
belief propagation
Complexity theory
Compressed sensing
Detection, estimation, filtering, equalization, prediction
Exact sciences and technology
expectation maximization algorithms
Gaussian mixture model
Information, signal and communications theory
Mathematical models
Maximization
Message passing
Noise
Noise measurement
Reconstruction
Recovery
Run time (computers)
Sampling, quantization
Sensors
Signal and communications theory
Signal, noise
Telecommunications and information theory
Vectors
Performance evaluation
State of the art
belief propagation
Mixture theory
Implementation
Gaussian mixture model
Mean square error
Credal approach
Gaussian process
Message passing
Minimax method
A priori estimation
Signal processing
expectation maximization algorithms
Bayes methods
EM algorithm
Compressed sensing
ISSN:1053-587X, 1941-0476
Online Access:Get full text
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Summary:When recovering a sparse signal from noisy compressive linear measurements, the distribution of the signal's non-zero coefficients can have a profound effect on recovery mean-squared error (MSE). If this distribution was a priori known, then one could use computationally efficient approximate message passing (AMP) techniques for nearly minimum MSE (MMSE) recovery. In practice, however, the distribution is unknown, motivating the use of robust algorithms like LASSO-which is nearly minimax optimal-at the cost of significantly larger MSE for non-least-favorable distributions. As an alternative, we propose an empirical-Bayesian technique that simultaneously learns the signal distribution while MMSE-recovering the signal-according to the learned distribution-using AMP. In particular, we model the non-zero distribution as a Gaussian mixture and learn its parameters through expectation maximization, using AMP to implement the expectation step. Numerical experiments on a wide range of signal classes confirm the state-of-the-art performance of our approach, in both reconstruction error and runtime, in the high-dimensional regime, for most (but not all) sensing operators.
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ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2013.2272287