Approximate Message Passing in Compressed Sensing Systems
Summary
Approximate Message Passing (AMP) algorithms have transformed the landscape of sparse signal recovery by providing low-complexity, iterative schemes that approach information-theoretic limits. Originating from belief-propagation concepts in graphical models, AMP applies a sequence of linear estimations and nonlinear denoising steps, each corrected by an Onsager term to cancel correlations introduced by previous iterations. This structure permits the derivation of a rigorous state evolution that predicts the mean-squared error at each step in the large-system limit. Variants such as orthogonal/vector AMP and convolutional AMP extend the basic framework to accommodate ill-conditioned measurement operators and structured priors. The versatility of AMP has been demonstrated across imaging, wireless communications and machine learning, offering both theoretical optimality under suitable randomness assumptions and practical robustness in finite-dimensional applications.
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Approximate Message Passing in Compressed Sensing Systems publication trend
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Technical terms
Approximate Message Passing (AMP): An iterative signal-recovery algorithm combining linear residual updates with nonlinear denoising and Onsager correction to achieve precise performance predictions in high dimensions.
State Evolution: A deterministic recursion that tracks the statistical behaviour of AMP estimates at each iteration in the large-system limit, enabling analysis of convergence and error.
Onsager Correction: A term added to AMP iterations to cancel spurious correlations arising from previous steps, ensuring the validity of state evolution.
Bayes-Optimal: Refers to algorithms or estimators that minimise the average reconstruction error under the true signal prior, achieving the best possible performance for given statistics.
Convolutional AMP (CAMP): A variant of AMP that replaces the Onsager term with a convolution over past messages to improve stability and convergence on ill-conditioned operators.
Sparsity: The property that a signal has only a small number of nonzero components relative to its ambient dimension, enabling recovery from undersampled measurements.
References
- Bayes-Optimal Convolutional AMP. IEEE Transactions on Information Theory (2021).
- On the Convergence of Orthogonal/Vector AMP: Long-Memory Message-Passing Strategy. IEEE Transactions on Information Theory (2022).
- Statistical-Physics-Based Reconstruction in Compressed Sensing. Physical Review X (2012).
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