Message Passing Algorithms for Compressed Sensing (0907.3574v1)

Abstract: Compressed sensing aims to undersample certain high-dimensional signals, yet accurately reconstruct them by exploiting signal characteristics. Accurate reconstruction is possible when the object to be recovered is sufficiently sparse in a known basis. Currently, the best known sparsity-undersampling tradeoff is achieved when reconstructing by convex optimization -- which is expensive in important large-scale applications. Fast iterative thresholding algorithms have been intensively studied as alternatives to convex optimization for large-scale problems. Unfortunately known fast algorithms offer substantially worse sparsity-undersampling tradeoffs than convex optimization. We introduce a simple costless modification to iterative thresholding making the sparsity-undersampling tradeoff of the new algorithms equivalent to that of the corresponding convex optimization procedures. The new iterative-thresholding algorithms are inspired by belief propagation in graphical models. Our empirical measurements of the sparsity-undersampling tradeoff for the new algorithms agree with theoretical calculations. We show that a state evolution formalism correctly derives the true sparsity-undersampling tradeoff. There is a surprising agreement between earlier calculations based on random convex polytopes and this new, apparently very different theoretical formalism.

• The paper introduces the AMP algorithm, an iterative thresholding technique that matches LP method accuracy while significantly reducing computational cost.
• It leverages state evolution formalism to predict performance, accurately capturing the sparsity-undersampling tradeoff in high-dimensional settings.
• Empirical results show that AMP achieves fast, scalable signal reconstruction, making it ideal for large-scale applications like MRI and spectroscopy.

Message Passing Algorithms for Compressed Sensing

Overview

The paper by Donoho, Maleki, and Montanari introduces an efficient iterative algorithm for reconstructing high-dimensional signals in the context of compressed sensing. This new algorithm, based on Approximate Message Passing (AMP), achieves performance comparable to that of conventional linear programming (LP) methods but with significantly reduced computational costs.

Key Contributions

1. Algorithm Introduction: The authors present the AMP algorithm, an iterative thresholding technique inspired by message passing methods in graphical models. The AMP algorithm is defined by two concise update equations, leveraging a computationally simple scalar thresholding function.
2. Theoretical Framework: Utilizing the state evolution (SE) formalism, the paper provides a robust theoretical framework predicting the behavior of the AMP algorithm. SE correctly anticipates the sparsity-undersampling tradeoff, matching the theoretical bounds established for LP-based methods.
3. Empirical Validation: The validity of the theoretical predictions is confirmed through extensive numerical simulations. The empirical phase transitions align closely with the SE predictions, demonstrating the accuracy of the simplified theoretical model.
4. Computational Efficiency: The paper outlines the dramatic computational advantages of AMP over traditional LP solvers, emphasizing its low per-iteration cost and feasibility for large-scale applications, such as medical imaging and spectroscopy.

Numerical Results

Key numerical results indicate that AMP achieves comparable performance to LP methods while running significantly faster. Specifically, in high-dimensional problems, AMP's computational cost scales favorably with the problem size, unlike LP solvers, which suffer from exorbitant computational demands.

Implications and Future Directions

Practical Implications:

• Scalability: AMP's computational efficiency makes it well-suited for applications involving large-scale data, such as MRI and compressed images, where traditional LP approaches are impractical.
• Accuracy: The ability of AMP to achieve the same sparsity-undersampling tradeoff as LP methods ensures that it can be adopted without compromising on reconstruction fidelity.

Theoretical Implications:

• State Evolution Formalism: The close agreement between SE predictions and empirical results establishes SE as a potent tool for analyzing iterative algorithms in compressed sensing.
• Algorithmic Design: The insights from AMP and SE could inspire the development of new, even more efficient algorithms for signal reconstruction in undersampled settings.

Speculative Future Developments:

• Enhanced Variants of AMP: Research could focus on refining the AMP algorithm or developing hybrid methods combining message-passing and other optimization techniques to further improve reconstruction performance or robustness.
• Expansion to Other Domains: The principles underlying AMP might be extended to other areas of computer science where high-dimensional data reconstruction is critical, such as genomics or network traffic analysis.

Conclusion

In summary, the paper presents a significant advancement in the field of compressed sensing by introducing the AMP algorithm, which combines the accuracy of LP methods with unprecedented computational efficiency. Theoretical predictions via state evolution are corroborated by empirical evidence, highlighting the robust and practical nature of this approach. The potential for AMP to influence future algorithmic developments and its applicability to large-scale data reconstruction tasks can make it a vital tool in the advancement of both theoretical and applied compressed sensing research.